Metropolitan Community College Contents Introduction 2 Unit 1 3 Rational Expressions........................................... 3 Quadratic Equations........................................... 9 Polynomial, Radical, Rational, and Absolute Value Equations.................... 12 Linear and Absolute Value Inequalities................................. 18 Functions................................................. 24 Extremum, Symmetry, Piecewise Functions, and the Difference Quotient.............. 27 Graphing Functions........................................... 34 Unit 2 41 Quadratic Functions........................................... 41 Polynomial Functions.......................................... 47 The Division Algorithm......................................... 50 The Fundamental Theorem of Algebra................................. 53 Rational Functions............................................ 58 Polynomial and Rational Inequalities.................................. 62 Unit 3 65 Operations on Functions......................................... 65 Inverse Functions............................................. 68 Exponential Functions.......................................... 72 Logarithmic Functions.......................................... 75 Properties of Logarithms......................................... 79 Exponential and Logarithmic Equations................................ 83 Unit 4 88 Circles................................................... 88 Ellipses.................................................. 92 Hyperbolas................................................ 96 1
Introduction The question I receive most often, regardless of the course, is, When am I ever going to use this? I think the question misses the point entirely. While I do not determine which classes students need to get their degree, I do think it is a good policy that students are required to take my course for more reasons than just my continued employment, which I support as well. If a student asked an English instructor why he or she had to read Willa Cather s My Ántonia, the instructor would not argue that understanding nineteenth century prairie life was essential to becoming a competent tax specialist or licensed nurse. The instructor would not argue that reading My Ántonia would benefit the student directly through a future application. Instead, the benefit of reading this beautiful piece of American literature is entirely intrinsic. The mere enjoyment and appreciation is enough to justify its place in a post-secondary education. Moreover, the results arrived to througout the course are as beautiful as any prose or poetry a student will encounter in his or her studies here at Metro or any other college. 2
Unit 1 Rational Expressions Domain of a Rational Expression A rational expression will be defined as long as the denominator does not equal zero. Example 1. State the domain of the rational expression. x x + 3 Example 2. State the domain of the rational expression. 2x + 3 3x 2 Example 3. State the domain of the rational expression. x 4 x 2 + 5x + 6 Example 4. State the domain of the rational expression. x 2 + 8x + 7 x 2 + 1 3
Example 5. Simplify. State any domain restrictions. 4x 8 x 2 Example 6. Simplify. State any domain restrictions. x 3 x 2 5x + 6 Example 7. Simplify. State any domain restrictions. x 2 14x + 49 x 2 6x 7 4
Example 8. Multiply. State any domain restrictions. 2x 3x + 1 3x2 5x 2 4x 2 + 8x Example 9. Multiply. State any domain restrictions. x 2 + x 6 4 x 2 x2 + 4x + 4 x 2 + 4x + 3 Example 10. Divide. State any domain restrictions. 2x 2 9x 5 2x 2 13x + 15 4x2 1 4x 2 8x + 3 5
Example 11. Add. State any domain restrictions. x 2 5x x 2 7x + 12 + 3x 3 x 2 7x + 12 Example 12. Add. State any domain restrictions. x 1 x 2 + x 20 + x + 3 x 2 5x + 4 6
Example 13. Subtract. State any domain restrictions. 4 x + 2 x 26 x 2 3x 10 Example 14. Simplify the complex rational expression. State any domain restrictions. 1 2 x 1 4 x 2 7
Example 15. Simplify the complex rational expression. State any domain restrictions. x x + 3 + 2 x + 2 x + 3 Example 16. Simplify the complex rational expression. State any domain restrictions. 1 + 6 x + 9 x 2 1 1 x 12 x 2 8
Quadratic Equations Definition: Quadratic Equation A quadratic equation is an equation that can be written as ax 2 + bx + c = 0 where a, b, and c are real numbers and a 0. Zero Factor Property If a b = 0, then a = 0 or b = 0. Example 1. Solve. x 2 5x + 6 = 0 Example 2. Solve. 3x(x 2) = 4(x + 1) + 4 Square Root Property If x 2 = a, then x = ± a. 9
Example 3. Solve. 3x 2 + 4 = 58 Example 4. Solve. (x 3) 2 = 4 Example 5. Solve. (2x 1) 2 = 5 10
Quadratic Formula For any quadratic equation ax 2 + bx + c = 0, x = b ± b 2 4ac 2a Example 6. Solve. 3x 2 5x 2 = 0 Example 7. Solve. x 2 3x 7 = 0 11
Polynomial, Radical, Rational, and Absolute Value Equations Example 1. Solve. x 3 16x = 0 Example 2. Solve. 8x 3 + 6x = 12x 2 + 9 Example 3. Solve. x + 1 = x + 13 12
Example 4. Solve. x 2 x + 3 1 = 2x Example 5. Solve. x 1 = 2x + 2 13
Example 6. Solve. (x 1) 2/3 = 4 Example 7. Solve. x 6 6x 3 + 9 = 0 14
Example 8. Solve. x 2 + 2x 1 15 = 0 Example 9. Solve. 8 (x 4) 2 6 x 4 + 1 = 0 15
Definition. Absolute Value. The absolute value of a real number x is the distance between 0 and x on the real number line. The absolute value of x is denoted by x. Example 10. Solve. x = 7 Observation 1 For any nonnegative value k, if x = k, then x = k or x = k. Example 11. Solve. x 3 = 2 Example 12. Solve. 3x + 5 = 8 16
Example 13. Solve. x + 2 = 3 Observation 2 For any negative value k, the equation x = k has no solution. Example 14. Solve. x + 9 3 = 1 Example 15. Solve. 2 3x + 2 + 1 = 0 17
Linear and Absolute Value Inequalities Definition: Union and Intersection Let A and B be sets. The union of A and B, denoted A B is the set of all elements that are members of A, or B, or both. The intersection of A and B, denoted A B is the set of all elements that are members of both A and B. Example 1. Let A = {1, 2, 3} and B = {2, 4, 6}. Determine both A B and A B. A B = A B = Example 2. Let B = {2, 4, 6} and C = {1, 3, 5}. Determine both B C and B C. B C = B C = Example 3. Let D = {x 0 < x < 4} and E = {x 2 < x < 6}. Determine both D E and D E. D E = D E = Interval Notation For any real numbers a and b, the following are sets written in interval notation. (a, b) = {x a < x < b} (a, b] = {x a < x b} [a, b) = {x a x < b} [a, b] = {x a x b} 18
Example 4. Write the following sets in interval notation. {x 3 x < 5} = {x 7 < x 10} = Example 5. Write the following sets in set-builder notation. (3, 8) = [ 2, 5] = Unbounded Intervals (a, ) = {x x > a} [ a, ) = {x x a} (, b) = {x x < b} (, b ] = {x x b} Example 6. Write the following sets in interval notation. {x x 2} = {x x < 2} = Example 7. Let A = (1, 4) and B = (2, 5). Determine both A B and A B. A B = A B = Example 8. Let B = (2, 5) and C = [3, 6]. Determine both B C and B C. B C = B C = 19
Example 9. Let D = (0, 4] and E = [5, 9). Determine both D E and D E. D E = D E = Example 10. Solve. 3x 7 < 5 Solution: Example 11. Solve. 2x 7 19 Solution: Example 12. Solve. 1 < 4x 3 11 Solution: 20
Example 13. Solve. 2 1 2x 3 3 Solution: Example 14. Solve. x < 4 Solution: Observation 3 For any nonnegative value k, the inequality x < k may be expressed as k < x < k. Similarly, for x k, we have k x k. Example 15. Solve. x > 4 Solution: Observation 4 For any nonnegative value k, the inequality x > k may be satisfied by either x > k or x < k. Similarly, for x k, we know x k or x k. 21
Example 16. Solve. x + 8 2 Solution: Example 17. Solve. 6x + 2 2 Solution: Example 18. Solve. 4 x < 8 Solution: 22
Example 19. Solve. 1 7x > 13 Solution: Example 20. Solve. x 3 > 2 Solution: Observation 5 For any negative value k, the inequality x > k holds for any value of x. Example 21. Solve. 3x + 2 < 5 Solution: Observation 6 For any negative value k, the inequality x < k has no solution. 23
Functions Definition: Relation A relation is a correspondence between two sets. Elements of the second set are called the range. Elements of the first set are called the domain. Definition: Function A function is a specific type of a relation where each element in the domain corresponds to exactly one element in the range. Example 1. Determine the domain and range of the following relation 1. Does the relation define a function? {(Joseph, turkey), (Joseph, roast beef), (Michael, ham)} Domain: Range: Function? Example 2. Determine the domain and range of the following relation. Does the relation define a function? {(1, 3), (2, 4), ( 1, 1)} Domain: Range: Function? Example 3. Determine the domain and range of the following relation. Does the relation define a function? {(3, 5), (4, 5), (5, 5)} Domain: Range: Function? 1 This relation relates math instructors and the sandwiches they enjoy. 24
Example 4. Determine whether the equation defines y as a function of x. x 2 + y = 1 Example 5. Determine whether the equation defines y as a function of x. x + y 2 = 1 Example 6. Determine whether the equation defines y as a function of x. x 2 + y 2 = 1 Example 7. Determine whether the equation defines y as a function of x. x 3 + y 3 = 1 25
Example 8. Evaluate the function for the given values. f(4) = f(x) = x 2 + 2x + 1 f( x) = f(x + h) = Example 9. Evaluate the function for the given values. f( 3) = f(x) = x 2 x 6 f( x) = f(x + h) = Example 10. Evaluate the function for the given values. f(2) = f(x) = x 3 3x 2 + 3x 1 f( x) = 26
Extremum, Symmetry, Piecewise Functions, and the Difference Quotient Increasing Functions, Decreasing Functions, Constant Functions Let f be a function and (a, b) be some interval in the domain of f. The function is called increasing over (a, b) if f(x) < f(y) for every x < y, decreasing over (a, b) if f(x) > f(y) for every x < y, and constant over (a, b) if f(x) = f(y) for every x and y (where a < x < y < b). Example 1. Determine over which intervals the function f is increasing, decreasing, or constant. Increasing: Decreasing: Constant: Relative Maximum: Relative Minimum: Domain: Range: Zeros of the function: 27
Example 2. Determine over which intervals the function f is increasing, decreasing, or constant. Increasing: Decreasing: Constant: Relative Maximum: Relative Minimum: Domain: Range: Example 3. Determine over which intervals the function f is increasing, decreasing, or constant. Increasing: Decreasing: Constant: 28
Even and Odd Functions A function f is called even if A function f is called odd if f( x) = f(x). f( x) = f(x). Example 4. Determine if f is even, odd, or neither. f(x) = x 2 4 Example 5. Determine if g is even, odd, or neither. g(x) = x 3 2x Example 6. Determine if h is even, odd, or neither. h(x) = (x 2) 2 29
Example 7. Evaluate the piecewise function. 2x + 8 if x 2 f(x) = x 2 if 2 < x 1 1 if x > 1 f( 3) = f( 1) = f(2) = f(4) = Example 8. Evaluate the piecewise function. f( 2) = f( 1) = f(1) = f(2) = Example 9. Graph the piecewise function. f(x) = f(x) = { x if x 0 x if x < 0 { x + 2 if x 0 1 if x > 0 y 3 2 1 x 3 2 1 1 2 3 1 30
Example 10. Graph the piecewise function. f(x) = { x if x 0 x if x < 0 y 3 2 1 x 3 2 1 1 2 3 1 Example 11. Graph the piecewise function. 2x + 8 if x 2 f(x) = x 2 if 2 < x 1 1 if x > 1 y 4 3 2 1 x 3 2 1 1 2 3 1 31
Difference Quotient For a function f(x), the difference quotient is f(x + h) f(x), h 0. h Example 12. Find the difference quotient of the given function. f(x) = 2x + 3 Example 13. Find the difference quotient of the given function. f(x) = 5x 6 32
Example 14. Find the difference quotient of the given function. f(x) = x 2 + 1 Example 15. Find the difference quotient of the given function. f(x) = x 2 4x 33
Graphing Functions The function f(x) = x 2 is called the square function. x f(x) 2 4 1 1 0 0 1 1 2 4 The function f(x) = x 3 is called the cube function. x f(x) 2 8 1 1 0 0 1 1 2 8 The function f(x) = x is called the square root function. x f(x) 0 0 1 1 4 2 9 3 34
The function f(x) = 3 x is called the cube root function. x f(x) 8 2 1 1 0 0 1 1 8 2 The function f(x) = x is called the absolute value function. x f(x) 2 2 1 1 0 0 1 1 2 2 Transformations of f(x) f(x) + c f(x) c f(x + c) f(x c) f(x) f( x) cf(x) f(cx) vertical shift up c units vertical shift down c units horizontal shift left c units horizontal shift right c units reflection over the x-axis reflection over the y-axis vertical stretch or compression by a factor of c horizontal compression or stretch by a factor of c 35
Example 1. Graph g(x) = x 2 + 1. Let f(x) = x 2. Note g(x) =f(x) + 1. x f(x) g(x) 2 4 5 1 1 2 0 0 1 1 1 2 2 4 5 Example 2. Graph g(x) = (x 2) 2. Let f(x) = x 2. Note g(x) =f(x 2). x 2 f(x 2) x g(x) 2 4 0 4 1 1 1 1 0 0 2 0 1 1 3 1 2 4 4 4 Example 3. Graph h(x) = (x + 4) 2 + 1. Let f(x) = x 2 and g(x) = (x + 4) 2. Note h(x) =g(x) + 1 =f(x + 4) + 1. x + 4 f(x + 4) x g(x) x h(x) 2 4 6 4 6 5 1 1 5 1 5 2 0 0 4 0 4 1 1 1 3 1 3 2 2 4 2 4 2 5 36
Example 4. Graph k(x) = (x 3) 2 1. Let f(x) = x 2, g(x) = (x 3) 2, and h(x) = (x 3) 2. Note k(x) =h(x) 1 = g(x) 1 = f(x 3) 1. x 3 f(x 3) x g(x) x h(x) x k(x) 2 4 1 4 1 4 1 5 1 1 2 1 2 1 2 2 0 0 3 0 3 0 3 1 1 1 4 1 4 1 4 2 2 4 5 4 5 4 5 5 Example 5. Graph g(x) = x 2. Let f(x) = x. Note g(x) =f(x) 2. x f(x) g(x) 0 0 2 1 1 1 4 2 0 9 3 1 37
Example 6. Graph g(x) = x 2. Let f(x) = x. Note g(x) =f(x 2). x 2 f(x) x g(x) 0 0 2 0 1 1 3 1 4 2 6 2 9 3 11 3 Example 7. Graph k(x) = x + 1 + 2. Let f(x) = x, g(x) = x + 1, and h(x) = x + 1. Note k(x) =h(x) + 2 = g(x) + 2 = f(x + 1) + 2. x + 1 f(x + 1) x g(x) x h(x) x k(x) 0 0 1 0 1 0 1 2 1 1 0 1 0 1 0 1 4 2 3 2 3 2 3 0 9 3 8 3 8 3 8 1 38
Example 8. Graph h(x) = (x 5) 3 2. Let f(x) = x 3 and g(x) = (x 5) 3. Note h(x) =g(x) 2 =f(x 5) 2. x 5 f(x 5) x g(x) x h(x) 2 8 3 8 3 10 1 1 4 1 4 3 0 0 5 0 5 2 1 1 6 1 6 1 2 8 7 8 7 6 Example 9. Graph k(x) = x + 2 + 1. Let f(x) = x, g(x) = x + 2, and h(x) = x + 2. Note k(x) =h(x) + 1 = g(x) + 1 = f(x + 2) + 1. x + 2 f(x 3) x g(x) x h(x) x k(x) 2 2 4 2 4 2 4 1 1 1 3 1 3 1 3 0 0 0 2 0 2 0 2 1 1 1 1 1 1 1 1 0 2 2 0 2 0 2 0 1 39
Example 10. The graph of the function f is given below. (a) Graph g(x) = f(x) 2. (b) Graph h(x) = f(x + 2). (c) Graph k(x) = f(x 1) + 2. 40
Unit 2 Quadratic Functions Vertex The vertex of a parabola is the point where the parabola achieves its minimum or maximum value. Example 1. Graph f(x) = x 2. Example 2. Graph f(x) = (x + 4) 2 1. Vertex: Example 3. Graph f(x) = 2(x 5) 2 + 3. Vertex: Vertex: 41
Example 4. Graph f(x) = 3(x 3) 2 1. Vertex: Standard and General Form of a Parabola A quadratic function is said to be in standard form if it is written as f(x) = a(x h) 2 + k. A quadratic function is said to be in general form if it is written as f(x) = ax 2 + bx + c. Example 5. Graph f(x) = x 2 + 8x + 15. Vertex: 42
Example 6. Graph f(x) = x 2 + 6x + 7. Vertex: Example 7. Graph f(x) = x 2 8x + 19. Vertex: 43
Example 8. Graph f(x) = 2x 2 4x 3. Vertex: Example 9. Graph f(x) = x 2 5x + 1. Vertex: 44
Example 10. Write the quadratic function f(x) = ax 2 + bx + c in standard form. Vertex: 45
Example 11. Evaluate the quadratic function f(x) = ax 2 + bx + c for x = b 2a. Vertex Formula For any quadratic function f(x) = ax 2 + bx + c, the vertex is located at ( b ( 2a, f b )). 2a Example 12. Graph f(x) = 3x 2 6x + 2. Example 13. Graph f(x) = 2x 2 7x + 5. 46
Polynomial Functions Polynomial Function A polynomial function is a function of the form f(x) = a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0, where n is a nonnegative integer and each a i is a real number. polynomial function is n and a n is called the leading coefficient. Assuming a n 0, the degree of the Example 1. Graph the following power functions. a. f(x) = x b. f(x) = x 2 c. f(x) = x 3 d. f(x) = x 4 e. f(x) = x 5 f. f(x) = x 6 g. f(x) = x 7 End Behavior The end behavior of a function is the value f(x) approaches as x approaches or as x approaches. Example 2. Identify the end behavior of each of the power functions in Example 1. Power Function x x f(x) = x f(x) f(x) f(x) = x 2 f(x) f(x) f(x) = x 3 f(x) f(x) f(x) = x 4 f(x) f(x) f(x) = x 5 f(x) f(x) f(x) = x 6 f(x) f(x) f(x) = x 7 f(x) f(x) 47
End Behavior of Any Polynomial Function The end behavior of any polynomial function is the same as the end-behavior of its highest degree term. Zero of a Function If f(c) = 0, then c is called a zero of the function. If c is a zero of a function, then (c, 0) is an x-intercept on the graph of the function. Example 3. Sketch a graph of f(x) = (x + 4)(x + 1)(x 2). Example 4. Sketch a graph of f(x) = (x + 4)(x + 1) 2 (x 2). 48
Example 5. Sketch a graph of f(x) = (x + 4) 2 (x + 1) 3 (x 2). Multiplicity of a Zero If (x c) n is a factor of f(x), but (x c) n+1 is not a factor of f(x), then c is a zero of multiplicity n. If c is a zero of multiplicity n, then: if n is odd, the graph crosses the x-axis, if n is even, the graph touches the x-axis, but does not cross. The Intermediate Value Theorem If f(x) is a polynomial function and a and b are real numbers with a < b, then if either f(a) < 0 < f(b), or f(b) < 0 < f(a), then there exists a real number c such that a < c < b and f(c) = 0. Example 6. Use the Intermediate Value Theorem to verify f(x) = x 3 + x + 1 has a zero on the closed interval [ 1, 0]. 49
The Division Algorithm The Division Algorithm Let p(x) be a polynomial of degree m and let d(x) be a nonzero polynomial of degree n where m n. Then there exists unique polynomials q(x) and r(x) such that p(x) = d(x) q(x) + r(x) where the degree of q(x) is m n and the degree of r(x) is less than n. The polynomial d(x) is called the divisor, q(x) is called the quotient, and r(x) is called the remainder. Example 1. Use long division to divide x2 3x 6. State the quotient, q(x), and remainder, r(x), x + 4 guaranteed by the Division Algorithm. Example 2. Use long division to divide x4 4x 3 + 6x 2 4x + 1. State the quotient, q(x), and remainder, x 1 r(x), guaranteed by the Division Algorithm. 50
Example 3. Use long division to divide x3 + x + 1. State the quotient, q(x), and remainder, r(x), guaranteed by the Division x + 1 Algorithm. Example 4. Use synthetic division to divide x3 3x 2 10x + 24. State the quotient, q(x), and remainder, r(x), guaranteed by the Division x 2 Algorithm. Example 5. Use synthetic division to divide x2 10x + 24. State the quotient, q(x), and remainder, x + 5 r(x), guaranteed by the Division Algorithm. 51
Example 6. Use synthetic division to divide x3 7x + 12. State the quotient, q(x), and remainder, r(x), x 3 guaranteed by the Division Algorithm. The Remainder Theorem Let p(x) be a polynomial. Then p(c) = r(x) where r(x) is the remainder guarenteed from the division algorithm with d(x) = x c. Example 7. Evaluate f(x) = x 4 3x 3 + 5x 2 7x + 8 for f(2) using the remainder theorem. Example 8. Evaluate f(x) = x 5 + 4x 3 9x + 2 for f( 1) using the remainder theorem. 52
The Fundamental Theorem of Algebra The Factor Theorem Let p(x) and d(x) be polynomials. p(x). If r(x) = 0 by the division algorithm, then d(x) is a factor of Example 1. Use the remainder theorem to verify that 3 is a zero of f(x) = x 3 3x 2 10x + 24. Then find all other zeros. Example 2. Use the remainder theorem to verify that 7 is a zero of f(x) = x 3 5x 2 13x 7. Then find all other zeros. 53
The Rational Zeros Theorem Let p(x) be a polynomial function with integer coefficients: p(x) = a n x n + a n 1 x n 1 +... + a 1 x + a 0. Then any rational zero of the polynomial will be of the form leading coefficient a n and constant term a 0. ± factor of a 0 factor of a n Example 3. List all possible rational zeros for f(x) = x 3 3x 2 10x + 24 given by the Rational Zeros Theorem. Example 4. List all possible rational zeros for f(x) = 2x 3 + 3x 2 32x + 15 given by the Rational Zeros Theorem. Descartes Rule of Signs Let p(x) be a polynomial function. The number of positive real zeros is equal to or less than by an even number the number of sign changes of p(x). The number of negative real zeros is equal to or less than by an even number the number of sign changes of p( x). 54
The Fundamental Theorem of Algebra Then p(x) has n complex zeros, including multi- Let p(x) be a polynomial function of degree n. plicities. Example 5. Find all zeros of the function f(x) = x 3 4x 2 + x + 6. Example 6. Find all zeros of the function f(x) = x 3 + 7x 2 + 16x + 12. 55
Example 7. Find all zeros of the function f(x) = x 4 4x 3 19x 2 + 46x 24. Example 8. Find all zeros of the function f(x) = x 4 x 3 2x 2 4x 24. 56
Complex Conjugate Theorem Let p(x) be a polynomial with real coefficients. If a + bi is a zero of the polynomial, then its complex conjugate a bi is also a zero of the polynomial. Example 9. Find a third degree polynomial f(x) with zeros of i and 3 such that f(0) = 3. Example 10. Find a third degree polynomial f(x) with zeros of 1 + i and 1 such that f(1) = 2. 57
Rational Functions Example 1. State the domain of the rational function. f(x) = x 1 x 2 x 6 Example 2. Graph the rational function. f(x) = 1 x Example 3. Graph the rational function. f(x) = 1 x 3 + 2 58
Vertical Asymptotes Let r(x) = n(x) d(x) asymptote. be a simplified rational function. If c is a zero of d(x), then x = c is a vertical Horizontal Asymptotes Let r(x) = n(x) d(x) be a rational function. 1. If the degree of the denominator, d(x), is greater than the degree of the numerator, n(x), then the line y = 0 is the horizontal asymptote. 2. If the degree of the denominator, d(x), is equal to the degree of the numerator, n(x), then the line y = ab is the horizontal asymptote, where a is the leading coefficient of n(x) and b is the leading coefficent of d(x). 3. If the degree of the denominator, d(x), is less than the degree of the numerator, n(x), then there is no horizontal asymptote. Holes Let r(x) = f(x) n(x) f(x) d(x) be a rational function. If c is a zero of f(x), then there is a hole at ( c, n(c) ). d(c) Example 4. Find any vertical or horizontal asymptotes. Identify any holes in the graph. f(x) = 3x x 2 9 59
Example 5. Find any vertical or horizontal asymptotes. Identify any holes in the graph. f(x) = 3x2 x 2 9 Example 6. Find any vertical or horizontal asymptotes. Identify any holes in the graph. f(x) = 3x3 x 2 9 Example 7. Find any vertical or horizontal asymptotes. Identify any holes in the graph. f(x) = 3x + 9 x 2 9 60
Example 8. Graph the rational function. f(x) = 1 x 2 + x 6 Example 9. Graph the rational function. f(x) = 2x x 2 4 Example 10. Graph the rational function. f(x) = 1 x 2 + 1 61
Polynomial and Rational Inequalities Example 1. Solve. x 2 7x + 12 = 0 Solution: Example 2. Solve. x 2 7x + 12 > 0 Solution: Example 3. Solve. x 2 + x 20 Solution: 62
Example 4. Solve. 4x 2 4x + 3 Solution: Example 5. Solve. x 3 x + 4 0 Solution: 63
Example 6. Solve. 2x 2 5x + 3 2 x 0 Solution: Example 7. Solve. x x + 4 2 Solution: 64
Unit 3 Operations on Functions Basic Operations on Functions Let f(x) and g(x) be functions. The following basic operations of addition, subtraction, multiplication, and division may be performed on the functions as follows: (f + g)(x) = f(x) + g(x) (f g)(x) = f(x) g(x) (f g)(x) = f(x) g(x) ( ) f (x) = f(x) g g(x) If the domain of f(x) is A and the domain of g(x) is B, then the domain of f + g, f g, and f g is A B. The domain of f/g is A B restricted for any x values such that g(x) = 0. Example 1. Let f(x) = 3x 2 and g(x) = x + 7. Find f + g, f g, f g, and f/g. State the domain of each function. 65
Composition of Functions Let f(x) and g(x) be functions. The composition of f and g, denoted f g, is given by (f g)(x) = f(g(x)). Example 2. Let f(x) = 3x 2 and g(x) = x + 7. Find f g and g f. State the domain of each function. 66
Example 3. Let f(x) = 1 and g(x) = x x + 7. Find f g and g f. State the domain of each function. x + 3 67
Inverse Functions Inverse Functions Two functions f and g are called inverse functions if (f g)(x) = (g f)(x) = x. Example 1. Verify that f and g are inverse functions. Graph both f and g. f(x) = 3x + 4 g(x) = x 4 3 Example 2. Verify that f and g are inverse functions. Graph both f and g. f(x) = x 3 + 2 g(x) = 3 x 2 68
Recall the definition of a function: One-to-one Function A function f is called one-to-one if each element in the range corresponds to exactly one element in the domain. If a function is one-to-one, then it has an inverse function. Example 3. Which of the following functions are one-to-one? 69
Example 4. Determine the inverse of the one-to-one function. State the domain and range of f and f 1. f(x) = 1 2 x 3 Example 5. Determine the inverse of the one-to-one function. State the domain and range of f and f 1. f(x) = x 2 3, x 0 70
Example 6. Determine the inverse of the one-to-one function. State the domain and range of f and f 1. f(x) = x 4 Example 7. Determine the inverse of the one-to-one function. State the domain and range of f and f 1. f(x) = 3 x + 2 71
Exponential Functions Exponential Function The function f(x) = b x, where b > 0 and b 1, is called an exponential function. Example 1. Graph f(x) = 2 x. State its domain and range. Example 2. Graph f(x) = 2 x + 3. State its domain and range. Example 3. Graph f(x) = 2 x+3. State its domain and range. 72
Example 4. Graph f(x) = 2 x. State its domain and range. Example 5. Graph f(x) = 3 x+2 4. State its domain and range. Natural Base Consider the expression ( 1 + 1 n) n for various values of n. See the table below. n ( ) 1 + 1 n n 1 2 10 2.59374 100 2.70481 1,000 2.71692 10,000 2.71815 100,000 2.71827 As n gets bigger, the expression ( 1 + 1 n) n gets bigger as well, but this sequence has an upper bound. This particular upper bound is called the natural base, e. e = 2.71828... Example 6. Graph f(x) = e x. State its domain and range. 73
Example 7. Graph f(x) = e x. State its domain and range. Example 8. Graph f(x) = 3e x + 1. State its domain and range. Example 9. Graph f(x) = e 2x 3. State its domain and range. 74
Logarithmic Functions Logarithm If b y = x, then log b x = y. The expression log b x is read the logarithm base b of x or log base b of x. Example 1. Write the following exponential equations as logarithmic equations. (a) 2 4 = 16 (b) 5 3 = 125 (c) 81 1 2 = 9 (d) 4 3 = 1 64 Example 2. Write the following logarithmic equations as exponential equations. (a) log 10 1000 = 3 (b) log 3 243 = 5 (c) log 27 3 = 1 3 (d) 3 = log 2 1 8 Example 3. Evaluate the following logarithms. (a) log 5 25 (a) log 10 10000 (a) log 3 1 (a) log 2 64 (a) log 36 1 6 75
Example 4. Find the inverse function of f(x) = 2 x. Example 5. Graph the function f(x) = 2 x and g(x) = log 2 x. 76
Example 6. Graph the function f(x) = log 3 x. State the domain and range. Natural and Common Logarithms The natural logaritm of x is denoted ln x and ln x = log e x. The common logaritm of x is denoted log x and log x = log 10 x. Example 7. Graph the function f(x) = log x. State the domain and range. Example 8. Graph the function f(x) = ln x. State the domain and range. 77
Example 9. Graph the function f(x) = log 2 (x 3). State the domain and range. Example 10. Graph the function f(x) = log 2 x 3. State the domain and range. 78
Properties of Logarithms Basic Properties of Logarithms 1. log b 1 = 0 2. log b b = 1 3. log b b x = x 4. b log b x = x Example 1. Evaluate the following expressions. (a) log 8 1 (b) log 4 4 (c) log 2 2 7 (d) 3 log 3 4 Example 2. Evaluate the following expressions. (a) ln 1 (b) log 10 (c) ln e 3 (d) e ln(2x) 79
Product Rule for Logarithms log b (M N) = log b M + log b N Proof. Quotient Rule for Logarithms ( ) M log b = log N b M log b N Proof. 80
Power Rule for Logarithms log b ( M N ) = N log b M Proof. Example 3. Expand the logarithmic expression. log 3 ( x 2 y 9z 3 ) Example 4. Condense the logarithmic expression. ln x + 5 ln y 3 ln z 81
Change of Base Formula log b M = log a M log a b Proof. Example 5. Approximate the logarithm. log 4 50 82
Exponential and Logarithmic Equations One-to-one Property for Exponential Functions If b x = b y, then x = y. Example 1. Solve the equation. 3 4x+1 = 81 Example 2. Solve the equation. 4 3x 1 = 8 x+5 83
One-to-one Property for Logarithmic Functions If x = y and x > 0, then log b x = log b y. Example 3. Solve the equation. 7 x 3 = 21 Example 4. Solve the equation. 4 2x 3 = 5 3x+4 84
Example 5. Solve the equation. e 2x+5 = 18 Example 6. Solve the equation. e 2x e x 6 = 0 85
Example 7. Solve the equation. log 3 (x + 3) = 2 Example 8. Solve the equation. ln(x + 7) = 4 86
Example 9. Solve the equation. log 2 (x 5) + log 2 (x + 2) = 3 Second One-to-one Property for Logarithmic Functions If log b x = log b y and x > 0 and y > 0, then x = y. Example 10. Solve the equation. ln(x + 3) ln x = ln 7 87
Unit 4 Circles Distance Formula The distance between any two points (x 1, y 1 ) and (x 2, y 2 ) is given by d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. y 2 (x 2, y 2 ) d y 2 y 1 y 1 (x 1, y 1 ) x 2 x 1 x 1 x 2 Example 1. Find the distance between the points ( 3, 4) and (3, 1). 88
Midpoint Formula The point halfway between two points (x 1, y 1 ) and (x 2, y 2 ) is called the midpoint by ( x1 + x 2, y ) 1 + y 2. 2 2 and is given y 2 (x 2, y 2 ) ( x1 + x 2 2, y ) 1 + y 2 2 y 1 (x 1, y 1 ) x 1 x 2 Example 2. Locate the midpoint between ( 3, 4) and (3, 1). 89
Example 3. Find the distance between (h, k) and any point (x, y). Circles A circle is the set of all points a fixed distance, called the radius, from a fixed point, called the center. Centered at (h, k) with radius r (x h) 2 + (y k) 2 = r 2 Example 4. Graph x 2 + y 2 = 1. 90
Example 5. Graph (x 3) 2 + (y + 1) 2 = 9. Example 6. Graph x 2 + 8x + y 2 4y 16 = 0. 91
Ellipses Ellipses Centered at (0, 0) An ellipse is the set of all points a fixed distance from two fixed point, called the foci. For both of the following equations, a > b and c 2 = a 2 b 2. Horizontal Major Axis x 2 a 2 + y2 b 2 = 1 (0, b) ( a, 0) (0, 0) (a, 0) (0, b) The foci are located at (c, 0) and ( c, 0). Vertical Major Axis x 2 b 2 + y2 a 2 = 1 (0, a) ( b, 0) (b, 0) (0, a) The foci are located at (0, c) and (0, c). 92
Example 1. Graph x2 25 + y2 9 = 1. Example 2. Graph x2 4 + y2 16 = 1. Example 3. Graph 9x 2 + 16y 2 = 144. Example 4. Write the equation of the ellipse that has vertices at (0, 6) and (0, 6) and foci at (0, 5) and (0, 5). 93
Ellipses Centered at (h, k) Horizontal Major Axis (x h) 2 (y k)2 a 2 + b 2 = 1 The vertices are located at (h + a, k) and (h a, k). The covertices are located (h, k + b) and (h, k b). The foci are located at (h + c, k) and (h c, k). Vertical Major Axis (x h) 2 (y k)2 b 2 + a 2 = 1 The vertices are located at (h, k + a) and (h, k a). The covertices are located (h + b, k) and (h b, k). The foci are located at (h, k + c) and (h, k c). Example 5. Graph (x + 4)2 20 + (y 2)2 36 = 1. Example 6. Graph x 2 + 9y 2 = 9. 94
Example 7. Graph 4x 2 8x + 9y 2 + 90y + 193 = 0. Example 8. Graph 4x 2 + 24x + y 2 10y + 57 = 0. 95
Hyperbolas Hyperbolas Centered at (0, 0) A hyberbola is the set of all points a fixed distance, when you subtract, from two fixed point, called the foci. For both of the following equations, c 2 = a 2 + b 2. Horizontal Transverse Axis x 2 a 2 y2 b 2 = 1 (0, b) ( c, 0) ( a, 0) (a, 0) (c, 0) (0, b) The vertices are located at (a, 0) and ( a, 0). The foci are located at (c, 0) and ( c, 0). Vertical Transverse Axis y 2 a 2 x2 b 2 = 1 (0, c) (0, a) ( b, 0) (b, 0) (0, a) (0, c) The vertices are located at (0, a) and (0, a). The foci are located at (0, c) and (0, c). 96
Example 1. Example 2. 97
Example 3. Example 4. 98
Example 5. Example 6. 99