Course Outline Functions/Graphing Solving Equations Applications Definitions of function, graph, domain, range, x- and y- intercepts, vertical line test(.-.) Linear functions (.-.5) -Parallel and perpendicular lines Piecewise Functions: Absolute value functions (.) Quadratic Functions: Completing the square, vertex, increasing/decreasing, maximum/minimum, graph movements (.,.5,.-.) Polynomial functions: Relative/Absolute Maximum/Minimum (4.-4.) Radical Functions: Graphing with movements Rational Functions: Horizontal/Vertical asymptotes (4.5) Algebra of Functions: Sums and Differences, Products and Quotients, Compositions, Difference Quotients (.-.) Inverse Functions (5.) Linear Equations Linear Inequalities (.6) Absolute Value Equations and Inequalities (.5) Quadratic Equations: Complex numbers, Quadratic formula (.-.) Polynomial Equations: Long/Synthetic Division (4.) Radical Equations (.4) Rational Equations (.4) Fahrenheit to Celsius conversion, Cost per hour, Average rate of change Maximum/minimum problems Finding Inverses Exponential Functions (5.) Exponential Equations (5.5) Interest, Growth/Decay (5.6) Logarithmic Functions (5.) Logarithmic Equations: Logarithm rules (5.4) Solving more complicated logarithmic/exponential equations (5.5)
Introduction to Functions and Graphing (Sections.-.) Definition. A function is a rule that assigns to each member of the first set (called the domain) exactly one member of a second set (called the range). x (input) f f(x) (output) Example. Determine whether or not each diagram represents a function. a a b b c c d d
Notation. Functions are usually given by formulas, such as: y = f(x) = g(x) = Example. Let f(x) = x + and g(x) = x + x +. Determine the outputs below. f( ) f(4) g( ) g() Example. Now let s try some that are a little harder. f(x) = 5x and g(x) = x + x. Determine the outputs below. f(t + 4) g(t + )
4 Definition. The graph of a function, f(x), is the picture obtained by plotting all ordered pairs of the form (x, f(x)), i.e. (input, output). Example. Determine if the following graphs correspond to functions. -9-8 - -6-5 -4 - - - - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9-9 -8 - -6-5 -4 - - - - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9-9 -8 - -6-5 -4 - - - - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9-9 -8 - -6-5 -4 - - - - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9 Theorem (Vertical Line Test). If it is possible for a vertical line to cross a graph more than once, then the graph is...
5 Definition. Given a function y = f(x): x-intercept(s) of the function y-intercept of the function domain of the function range of the function Interval Notation:
6 Example. Graph the following functions using a table. Then use the graph to estimate the x- and y-intercepts, as well as the domain and range. f(x) = x + 4x + 5-9-8--6-5-4--- - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9 h(x) = x -9-8--6-5-4--- - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9
Example. You can also use a table to draw graphs of equations that are not functions. After you have drawn the graph, estimate the x- and y-intercepts, as well as the domain and range. y = ± x -9-8--6-5-4--- - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9 x + y = 4-9-8--6-5-4--- - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9
8 Example. Graph the following on your graphing calculator. Using the graph, estimate the x- and y-intercepts, as well as the domain and range. y = x y = x 4 x x-intercept(s) x-intercept(s) y-intercept y-intercept Domain Domain Range Range y = x + 5 y = x 4 x-intercept(s) x-intercept(s) y-intercept y-intercept Domain Domain Range Range Since we can t always rely on the graph to be able to determine the x-/y-intercepts and the domain/range exactly, we will have to learn algebraic methods as well.
9 Linear Functions and Linear Equations (Sections.-.5) Definition. A function f is called a linear function if it can be written in the form where m and b are constants. Example. Determine whether each of the following is a linear function. y x = 5 y 4x + = 0 y x = 0 (8x y + 4) = 0
0 Remark. The two constants m and b represent important parts of the graph. The constant b is The constant m is called the slope of the line and is a measure of the steepness of the line where (x, y ) and (x, y ) are two points on the line. (x, y ) (x, y ) y y (rise) x x (run)
Example. Graph the following linear functions without using a table. f(x) = x y = x + -9-8 - -6-5 -4 - - - - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9-9 -8 - -6-5 -4 - - - - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9 x + y = 4 g(x) = -9-8 - -6-5 -4 - - - - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9-9 -8 - -6-5 -4 - - - - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9
Example. Find the slope of the line through the two given points. (, ) and (, 5) (6, 8) and (, 8) (, ) and (5, ) (, 0) and (, 8) (, 4) and (, 4) (0, ) and (8, )
Remark. Horizontal lines: y = b Are linear functions Vertical lines: x = a Are NOT linear functions Example. Graph the following lines. y = y = 5-9 -8 - -6-5 -4 - - - - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9-9 -8 - -6-5 -4 - - - - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9 x = 4 x = -9-8 - -6-5 -4 - - - - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9-9 -8 - -6-5 -4 - - - - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9 Note: The slope of a horizontal line is. The slope of a vertical line is.
4 Remark. To find the equation of a line, you can use one of the following forms of a line. However, your answer should always be given in slope-intercept form. Slope-Intercept Form Point-Slope Form y = mx + b y y = m(x x ) Example. Find the equation of the line through (, 5) having slope m =. Example. Find the equation of the line given a point on the line and the slope. (, 4) and m = (0, 5) and m =
5 Example. Find the equation of the line through (, ) and (, 5). Example. Find the equation of the line through the given points. (, 5) and (0, 4) (, 5) and (, 8) (, 6) and (, 6) (4, ) and (, 5)
6 Remark. To find x- and y-intercepts algebraically: Set x = 0 for y-intercept Note: For linear equations in the form y = mx + b, the y-intercept is just (0, b) Set y = 0 for x-intercept(s) Example. Find the x- and y-intercepts for the graph of each equation. y = x + y = x + x-intercept(s) y-intercept x-intercept(s) y-intercept x = y = 5 x-intercept(s) y-intercept x-intercept(s) y-intercept
Applications of Linear Functions Average Rate of Change (Section.) Remark. The average rate of change between any two data points on a graph is the slope of the line passing through the two points. Example. Kevin s savings account balance changed from $40 in January to $450 in April. Find the average rate of change per month. Example. A plane left Chicago at 8:00am. At :00pm, the plane landed in Los Angeles, which is 500 miles away. What was the average speed of the plane for the trip? Example. Find the average rate of change for each function on the given interval. f(x) = x + x x on [, ] f(x) = x on [4, ]
8 Applications of Linear Functions Modeling (Section.4-.5) Example. Given the following information, determine a formula to convert degrees Celsius to degrees Fahrenheit. Water freezes at 0 Celsius and Fahrenheit. Water boils at 00 Celsius and Fahrenheit. Example. A taxi service charges a $.5 pickup fee and an additional $.5 per mile. If the cab fare was $.60, how many miles was the cab ride?
9 Parallel and Perpendicular Lines (Section.4) Definition. Two distinct lines are called parallel iff () they are both vertical lines, or () they have the same slope Example. Determine if the two lines are parallel. 6x y = 4y = 8x 4y 8 = x y + 4x = Example. Find the equation of the line through the given point and parallel to the given line. (0, ), y = x + (9, ), y = 4
0 (, 5), y = x + (5, 4), x = Definition. Two distinct lines are called perpendicular iff () one is vertical and one is horizontal, or () their two slopes, m and m, satisfy m m = Example. Determine if the two lines are perpendicular. y = x + 4 4y + x + = 0 5y = x 5x y = 4
Example. Find the equation of the line through the given point and perpendicular to the given line. (0, ), y = x + (9, ), y = 4 (, 5), y = x + (5, 4), x =
Intersection Points and Zeros (Section.5) Definition. An intersection point of two functions is a point (x, y) where the graphs of each function cross (i.e. it is a point that is in common to both graphs). Example. Find the intersection point (if any) between the two given lines. y = x + and y = 5x 6 y = x 5 and y = x + 8 y = 4 x + and y = 4 x 5 y = 5 and x = Example. A private plane leaves Midway Airport and flies due east at a speed of 80 km/h. Two hours later, a jet leaves Midway and flies due east at a speed of 900 km/h. After how many hours will the jet overtake the private plane?
Definition. Zeros of a function are the values of x that make the function equal 0 (also the x-coordinate of the x-intercept). Example. Find the zero(s) of each function (if any). f(x) = 8x + 4 g(x) = 5 g(x) = x f(x) = 4x + Example. A plane is descending from a height of 8,000 feet at a constant rate of 00 feet per minute. After how long will the plane have landed on the ground?
4 Linear Inequalities (Section.6) Rules: The sign stays the same unless you mutliply/divide by a negative, then the sign flips. Example. Solve each inequality. Then graph the solution set, and then write the solution set in interval notation. x + 5 6x 6 + x > 4x + 8 5x + < 0 4 < x + 4 Example. Macho Movers charges $00 plus $45 per hour to move a household across town. Brute Force Movers charges a flat fee of $65 per hour for the same move. For what lengths of time does it cost less to hire Brute Force Movers?
5 Piecewise Defined Functions (Section.) These are just functions whose rules are defined in different parts/pieces. -9-8 - -6-5 -4 - - - - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9 Example. Determine the function values. (a) f(x) = f() { x +, x > x, x x + 5, x 0 (b) g(x) = x +, 0 < x 5, x > 5 g( ) g(0) f() g() f( 4) g(8)
6 Absolute Value Function (Sections. and.5) { x, x < 0 Definition. h(x) = is the absolute value function, denoted by h(x) = x. x, x 0 Remark. To solve absolute value equations: = a No solution if a < 0 Equivalent to = a or = a if a 0 Example. Solve each of the following. x = x + = 5 x + = 6 x + 6 = 5
Remark. To solve absolute value inequalities when a > 0: < a means a < < a (Also true for ) > a means > a or < a (Also true for ) Example. Solve each of the following. Write your answer in interval notation. x < 4 x 0.5 0. x 8 6 4x > 8
8 Quadratic Functions (Sections. and.) Definition. Quadratic functions are functions that can be written in the form f(x) = ax + bx + c, a 0 where a, b, and c are real numbers. Example. Use a table to sketch the graph of f(x) = x. -9-8 - -6-5 -4 - - - - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9 Now use your graphing calculator to graph each of the following and describe how it is different from the graph of f(x) = x : y = (x ) y = x y = x y = (x + ) y = x + y = x y = (x 5) y = x 5 y = 5x y = (x + 5) y = x + 5 y = 5 x For c > 0: y = (x c) y = x c y = cx y = (x + c) y = x + c y = cx Also: y = x
9 Example. Describe how the graph of g(x) could be obtained from the graph of f(x) = x. g(x) = (x 4) g(x) = (x + 6) g(x) = (x + ) + 9 g(x) = (x + ) g(x) = (x 5) + 6 g(x) = (x 5 ) + 4 Remark. Every quadratic function (i.e. function of the form f(x) = ax + bx + c where a 0), can be written in the form: f(x) = a(x h) + k. This is called the for a quadratic function and (h, k) is the.
0 Definition. () A function f is called increasing on an open interval I, if for all a, b I, a < b implies f(a) < f(b). (Rises as you go to the right) () A function f is called decreasing on an open interval I, if for all a, b I, a < b implies f(a) > f(b). (Falls as you go to the right) () A function f is called constant on I, if f(a) = f(b) for all a, b I. Example. For each of the functions below, determine the intervals of increase and decrease. f(x) = (x + ) Increasing: g(x) = (x 6) Increasing: Decreasing: Decreasing: -9-8--6-5-4--- - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9-9-8--6-5-4--- - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9
f(x) = x + 4 Increasing: f(x) = (x + 4) + Increasing: Decreasing: Decreasing: -9-8 - -6-5 -4 - - - - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9-9 -8 - -6-5 -4 - - - - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9 Completing the Square If every quadratic function f(x) = ax + bx + c where a 0 can be written in the form f(x) = a(x h) + k, how do you get it in that form? f(x) = x + 4x Method
Example. Complete the square to write each quadratic function in the form f(x) = (x h) + k. Determine the vertex of the parabola, the axis of symmetry, and the intervals of increase/decrease. f(x) = x 6x + 0 f(x) = x + 0x + 8 Remark. Notice that the coordinates of the vertex for f(x) = x + bx + c are. Example. Write the following quadratic functions in the form f(x) = (x h) + k. f(x) = x + x f(x) = x + 8x + 4
What happens if a 0 in f(x) = ax + bx + c? How do you complete the square to write it in the form f(x) = a(x h) + k? f(x) = x + 8x Method Example. Complete the square to write each quadratic function in the form f(x) = (x h) + k. Determine the vertex of the parabola, the axis of symmetry, and the intervals of increase/decrease. f(x) = 4x + 8x f(x) = x 9x + Remark. Notice that the coordinates of the vertex for are. f(x) = ax + bx + c
4 Example. Write the following quadratic functions in the form f(x) = a(x h) + k. f(x) = x x f(x) = x + 5x + 4 Definition. Let f(x) be a function. Then f(c) is...... called a relative (local) maximum if there exists an open interval I containing c such that f(c) > f(x), for all x in I where x c.... called a relative (local) minimum if there exists an open interval I containing c such that f(c) < f(x), for all x in I where x c.
Remark. A function can have several relative maximums and several relative minimums. Here are some examples: 5 Maximum and Minimum Applications (Section.) Example. A model rocket is launched into the air with an initial velocity of 90 ft/sec from the top of a building that is 40 feet high. The rocket s height, in feet, t seconds after it has been launched is given by the function: s(t) = 6t + 90t + 40. Determine the maximum height that the rocket will reach along with the time when the rocket will reach that maximum height.
6 Example. A manufacturer of dog houses has determined that when x hundred wooden doghouses are built, the average cost (in hundreds of dollars) per doghouse is given by C(x) = 0.x 0.x +.65. How many doghouses should be built in order to minimize the average cost per doghouse? Example. In business: P (x) = R(x) C(x) Profit Revenue Cost (in $) (in $) (in $) where x is the number of units sold. Find the maximum profit, along with how many units must be sold when: R(x) = 50 0.5x C(x) = 0x +
Example. When Mentos are added to Diet Coke, the Diet Coke is propelled upward from ground level at a rate of feet per second. The height of the Diet Coke (in feet) after t seconds is given by the function s(t) = 6t + t. Determine the maximum height of the Diet Coke. Example. A landscaper has enough stone to enclose a rectangular koi pond next to an existing garden wall of a house with 4 feet of stone wall. If the garden wall forms one side of the rectangle, what is the maximum area that the landscaper can enclose? What dimensions should the koi pond be?
8 Quadratic Equations (Section.) Definition. A quadratic equation is an equation that can be written in the form ax + bx + c = 0 where a 0. Example. Determine if the following equations are linear, quadratic, or neither. 4x = 8 9x x + 4 = x x 8 = 4x x + x = 4x 5x = 8 x = 5 x How do you solve quadratic equations? Step # : Move everything to one side of the equation. Step # : Try to factor. Step # : If you can t seem to factor the equation, use the quadratic formula.
9 The solutions of ax + bx + c = 0 where a 0 are given by: x = b ± b 4ac. a Where does the quadratic formula come from?
40 Example. Solve the following equations. x x = 0 x = 8x 9 4x = 0 y + 8y = 5
4 Example. (Continued from previous page) Solve the following equations. w + 9 = 6w x 4x + 5 = 0 How many solutions does ax + bx + c = 0 have where a 0? Real Solutions and the Discriminant Overall Solutions
4 Applications with Quadratic Equations (Section.) Example. The number of U.S. forces in Afghanistan decreased to approximately 4,000 in 04 from a high of about 00,000 in 00. The amount of U.S. funding for Afghan security forces also decreased during this period. The function f(x) =.x + 5.56x + 5.5 can be used to estimate the amount of U.S. funding for Afghan security forces, in billions of dollars, x years after 009. In what year was the amount of U.S. funding for Arghan security forces about $0.5 billion? (Source: U.S. Department of Defense; Brookings Institution; International Security Assistance Force; ESRI ). Example. A stone is thrown directly upward from a height of 0 feet with an initial velocity of 60 feet per second. The height of the stone, in feet, t seconds after it has been thrown is given by the function s(t) = 6t + 60t + 0. Determine the total number of seconds the stone was in the air.
Quadratic Type Equations (Section.) Sometimes an equation is not a quadratic equation, but using a substitution of variables we can make it look like a quadratic equation. Consider: x 4 5x + 4 = 0 4 Example. Solve the following quadratic type equations. x 4 x + = 0 x 4 8x = 9
44 Polynomial Functions and Graphs (Section 4.-4.) Definition. A polynomial function is a function of the form where the coefficients a n, a n,... a, a 0 are real numbers and the exponents are whole numbers. The degree of the polynomial P (x) is n, and the leading coefficient of P (x) is a n. Example. Determine the degree, leading coefficient, domain, and range of each polynomial function. Function Degree Leading Domain Range Tiny Coefficient Sketch f(x) = 5 f(x) = x + f(x) = x 6x + 5 f(x) = x + 4x f(x) = x + x f(x) = x 4 + x f(x) = x 5 + 4x + x Can you describe any aspects of the shape of the graph of f(x) based on its degree? Even Degree Odd Degree
45 Remark (The Leading-Term Test). For the polynomial P (x) = a n x n + a n x n + + a x + a 0 the behavior of the graph as x (as the graph goes to the far right) and as x (as the graph goes to the far left): n a n > 0 a n < 0 Example. Determine the end behavior of each polynomial. f(x) = 4x 5x + f(x) = x + 4x 6 x 5 + x f(x) = x 0 x 9 + 6x 4 + x 9 f(x) = x 5 + x 4 + x x f(x) = 4x 8 4x 6 + x 5 + 8 f(x) = x 00 x + 8x 50
46 Solving Polynomial Equations (Sections 4.-4.) Example. Solve the following equations. x x 8x = 0 x 4 + 4x 45 = 0 But what if we couldn t see how to factor easily? If we can guess/find one root (i.e. solution) to the equation, we can use that to help us factor using the following relationship: x = c is a solution to f(x) = 0 (x c) is a factor of f(x) To use this relationship, we need to know how to do polynomial long division. Example. x = is a zero of f(x) = x x 9x + 8. So (x ) is a factor:
4 Example. Use polynomial long division to factor f(x), given a zero of f(x). f(x) = x + x 5x 6, x = f(x) = x 4 6x + x + 4x 0, x = Polynomial long division takes up a lot of space... and since we are only dividing by (x c) we can use a process called synthetic division that works in these cases. Example. Let s consider the problem f(x) = x 4 6x + x + 4x 0 with the zero x = from the previous example.
48 Example. Use synthetic division to factor the following functions, given a zero of the function. f(x) = x x 6x + 8, x = f(x) = x 4 x + 9x + x 54, x = How can we use this to sketch a graph of a polynomial? Consider f(x) = x x 6x + 8 (from the previous example). Step #: Find a zero by guess & check. Step #: Use synthetic division to factor. Step #: Completely factor f(x). Step #4: Plot all x-intercepts. -9-8 - -6-5 -4 - - - - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9 Step #5: Use a sign chart to sketch the graph.
49 Example. Sketch the graph of each function. f(x) = x x + 6-9 -8 - -6-5 -4 - - - - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9 f(x) = x 4 + x + x 4x 4-9 -8 - -6-5 -4 - - - - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9
50 Polynomial Inequalities (Section 4.6) Example. Solve the following inequalities. x 4x 5 > 0 x + x 5 x + x < x x 9x + x
5 Radical Functions (Section.4) Definition. Radical functions are functions that involve Example. Use a table to sketch the graph of f(x) = x. n -9-8 - -6-5 -4 - - - - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9 Now use your graphing calculator to graph each of the following and describe how it is different from the graph of f(x) = x: y = x y = x y = x y = x + y = x + y = x y = x y = x Remark. So just like we saw for f(x) = x...
5 Example. Describe how the graph of g(x) could be obtained from the graph of f(x) = x. g(x) = x 4 + g(x) = x + g(x) = x 5 8 g(x) = 4 x g(x) = x + 6 g(x) = 0.9 x Example. Given the graph of y = f(x), sketch the graph of each. -9-8--6-5-4--- y = f(x ) y = f(x) + 4 y = f( x) - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9-9-8--6-5-4--- - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9-9-8--6-5-4--- - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9
5 Example. Determine the domain and range for each of the following functions below. Recall: The domain and range of f(x) = x are f(x) = x + 4 f(x) = x + 9 f(x) = x + 4 f(x) = x + 4 f(x) = x f(x) = x +
54 Example. Simplify the following. Solving Radical Equations (Section.4) 64 8 4 8 5 4 How do you use your calculator to help with things like 4 5? Example. Use your calculator to round the following to three decimal places. 8 4 6 5 9
55 How do we solve an equation like x = 4? Example. Solve each of the following equations. 4x + = y + 4 = 0 5 x + 4 = 6x 9 + 8 = 5
56 How do we solve an equation like x + + = x? Example. Solve each of the following equations. x + 4 = x + x + 5 = x
5 Graphs of Rational Functions (Section 4.5) Definition. A rational function if a function of the form: where p(x) and q(x) are polynomials and q(x) 0. Example. Graph the following functions on your graphing calculator and determine the domain. f(x) = f(x) = x + x + 5 x x 6 Remark. The domain for a rational function f(x) = p(x) q(x) is: Example. Determine the domain for each of the following functions. x f(x) = f(x) = x 4x x + x 8 x x
58 Graph f(x) = x 4x x x on your graphing calculator: -9-8 - -6-5 -4 - - - - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9 Remark. To find vertical asymptotes for a rational function f(x) = p(x) q(x) : () Reduce f(x) by canceling out any common factors between p(x) and q(x). () Set the new denominator equal to zero. () The vertical asymptotes correspond to the answers found above. Example. Determine the vertical asymptotes for each of the following rational functions. f(x) = x x f(x) = x + 6 x + x 8 x + x 8
59 How do we solve an equation like x 8 Solving Rational Equations (Section.4) + x = 0?... or how would you solve an equation like x x + = 5 x?
60 Example. Solve the following equations. x + 6 x = 5 + x = + x x = x + y + 5 y + 5y + y + 4 y + 5 = y + y
More About Graphs of Rational Functions (Section 4.5) Previously, we learned how to find the domain, vertical asymptotes, and horizontal asymptotes for a rational function. This information told us some keys as to what the graph of the function would look like. Now, we review how to find x- and y-intercepts. To graph a rational function accurately, you would need to do all of these things and then to create a table to find additional points. Example. Find the x- and y-intercepts for following functions. f(x) = x + x 0 g(x) = x 4x x x x x 6
6 Compositions of Functions (Section.) In the beginning of the semester, we did problems like this one: For f(x) = x and g(x) = x + 4x find the following: f(4 + h) g( + h) This leads into the composition of functions. For the two functions f(x) and g(x), the composition functions f g and g f are defined as: Example. Consider the functions f(x) = x + and g(x) = x + 4. Find: (f g)(x) (g f)(x) Remark. Notice that:
6 Example. Let f(x) = x, g(x) = x +, and h(x) = x + 8. Find: (g f)(x) (f f)(x) (f h)(x) (h g)(x) It is very important in Calculus that you can do compositions of functions, because the derivative is defined using compositions of functions via the difference quotient. For a function f(x), the difference quotient of f(x) is: f(x + h) f(x) h Let s find the difference quotient for f(x) = 4x.
64 Example. Find the difference quotient for each function. f(x) = x + g(x) = x x Example. Consider f(x) = x and g(x) = x +. Find (f g)(x) and (g f)(x)
65 Inverses Functions (Section 5.) In the previous example, f(x) = x and g(x) = x + are inverses: Definition. Two functions f and g are called inverses if Example. Determine if the given two functions are inverses of each other. f(x) = x + and g(x) = x + f(x) = x and g(x) = x x +
66 So when does a function have an inverse? And then how do we find the inverse? Example. Determine the inverse of f(x) (if it exists). f(x) = x 4x + f(x) = x + 5
6 Example. Determine the inverse of the given function (if it exists). f(x) = g(x) = 5 x + x We know want to look at two different types of functions, called logarithmic functions and exponential functions. It will turn out that these two types of functions are inverses of each other, i.e. the inverse of a logarithmic function will be an exponential function and the inverse of an exponential function will be a logarithmic function. Definition. The function Exponential Functions (Section 5.) x is a real number, a > 0, and a, is called the exponential function with base a. Example. Consider the exponential functions Find the following output values. f(x) = x and g(x) = ( ) x and h(x) = e x. f( ) = g( ) = h( ) = f(0) = g(0) = h(0) = f() = g() = h() =
68 Let s draw the graphs of f(x) = x and f(x) = -9-8--6-5-4--- - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9 ( ) x : Now, let s draw the graphs of f(x) = x and f(x) = -9-8--6-5-4--- - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9 So graphs of exponential functions look like: -9-8--6-5-4--- ( ) x : - - - -4-5 -6 - -8-9 -9-8--6-5-4--- - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9 9 8 6 5 4 4 5 6 8 9 -
69 What does log a x mean???? Logarithms (Section 5.) Example. Calculate the following. log 8 log 5 log 00.00 log 8 log ( 8 ) log 4 6 Definition. The function is the logarithmic function having base a, where a > 0 is a real number. Two bases that occur often have special notations:
0 Example. Compute the following. Round your answers to two decimal places. log ln log 0 ln 0 Your calculator has buttons for ln x and log x, but how do you calculate other base logarithms? Example. Compute the following. Round your answers to two decimal places. log 5 log 6 log 6 0 You can convert between logarithmic and exponential equations using the following:
Example. Convert the following logarithmic equations to exponential equations. log 5 5 = log = 0.845 ln 40 = x Example. Convert the following exponential equations to logarithmic equations. e = t 5 = 0 = 000 5 Graphs of Inverse Functions (Section 5.) Let s compare the graphs of two inverse functions. Graph f(x) = x and f (x) = x + on the same coordinate axes: -9-8 - -6-5 -4 - - - - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9
Let s try another pair of inverse functions. Graph f(x) = x + and f (x) = x on the same coordinate axes: -9-8 - -6-5 -4 - - - - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9 Since f(x) = a x and f (x) = log a (x), we can use the graph of f(x) = a x to graph f (x) = log a (x). Example. Sketch the graph of each logarithmic equation. y = log x y = log x -9-8 - -6-5 -4 - - - - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9-9 -8 - -6-5 -4 - - - - - - -4-5 -6 - -8-9 9 8 6 5 4 4 5 6 8 9
Application to Compound Interest (Section 5.) If P dollars are invested at an interest rate r (written as a decimal), compounded n times per year, then the amount A of money in the account after t years is given by: ( A = P + r ) nt n Example. If $500 is invested at a rate of.5%, compounded monthly, how much money will be in the account after 5 years? Example. If $00 is invested at a rate of.5%, compounded semiannually, how much money will be in the account after years? As the number of times you compound interest per year increases, the amount in the account after t years also increases. Consider investing $00 at a rate of % for a time period of 5 years. So ( A = 00 +.0 ) 5n n
4 If interest is compounded continuously, then the amount A in the account after t years is: A = P e rt Example. If $500 is invested at a rate of.5%, compounded continuously, how much money will be in the account after 5 years? Example. If $00 is invested at a rate of.5%, compoundedcontinuously, how much money will be in the account after years? Solving Exponential and Logarithmic Equations (Section 5.5) Logarithm and exponential functions are inverses, so: log a (a x ) = x and a log a (x) = x Example. Solve the following equations: x+ = 5 log 5 (x ) =
5 Example. Solve the following equations: 5 4x = 5 log 5 (8 x) = x = 40 ln x = Properties of logarithms: log a M + log a N = log a MN log a M log a N = log a M N p log a M = log a M p
6 So, how do we solve logarithmic equations like log 8 (x + ) log 8 x = Example. Solve the following equations: log (x + ) + log (x ) = log x + log(x + 4) = log
One more example... solve ln(x + 8) + ln(x ) = ln x Let s take a look back at solving exponential equations. Example. Solve the following equations: 000e 0.09t = 5000 50 (.8) x = 0
8 Example. Suppose that $,500 is invested at an interest rate of.5%, compounded quarterly. How long would it take for the amount of money in the account to become $,000? Example. Suppose that $5,000 is invested at an interest rate of 5.4%, compounded continuously. How long would it take for the amount of money to double?