# Review all the activities leading to Midterm 3. Review all the problems in the previous online homework sets (8+9+10).

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1 MA109, Activity 34: Review (Sections ) Date: Objective: Additional Assignments: To prepare for Midterm 3, make sure that you can solve the types of problems listed in Activities 33 and 34. At least one sample problem is listed by each type of problem. The sample problems are found in your textbook and are reproduced here for your convenience. Make sure that you can solve each of the sample problems without consulting your notes or examples in the textbook. Review all the activities leading to Midterm 3. Review all the problems in the previous online homework sets (8+9+10). Section 3.6: Combining Functions Find the sum, difference, product, and quotient of two functions and state their domains. (Numbers 1 and 3) Understand function composition. (Numbers 17 and 21) Find the domain of a composite function. (Number 33) Problems 1 and 3: Find f + g, f g, fg, and f/g and their domains. f(x) = x 2, g(x) = x + 2 f(x) = 1 + x 2, g(x) = 1 x Problems 17 and 21: Use f(x) = 3x 5 and g(x) = 2 x 2 to evaluate the expressions. f(g(0)) g(f(0)) (f g)(x) (g f)(x) Problem 33: Find the functions f g, g f, f f, and g g and their domains for f(x) = 1 x and g(x) = 2x+4. 82

2 Section 3.7: One-to-One Functions and Their Inverses Know the definition of a one-to-one function. (See page 280 of your textbook.) Given the graph of a function, use the horizontal line test to determine whether the function is one-to-one. (Numbers 1 and 3) Understand the definition of an inverse function. (Numbers 17 and 19) Understand the Property of Inverse functions on page 282 of your textbook. (Number 23) Given a one-to-one function, find its inverse function. (Numbers 33 and 37) Given the graph of a one-to-one function, sketch the graph of its inverse function. (Number 65) Problems 1 and 3: Using the graphs for Problems 1 and 3 on page 286 of your textbook, determine whether each function is one-to-one. Problem 17: Assume f is a one-to-one function. If f(2) = 7, find f 1 (7). If f 1 (3) = 1, find f( 1). Problem 19: If f(x) = 5 2x, find f 1 (3). (You may assume that f is a one-to-one function.) Problem 23: Use the Property of Inverse Functions to show that f(x) = 2x 5 and g(x) = x of each other. are inverses Problems 33 and 37: Find the inverse function of f. f(x) = 4x + 7 f(x) = 1 x + 2 Problem 65: Using the graph of f for Problem 65 on page 287 of your textbook, sketch the graph of f 1. 83

3 Section 4.1: Polynomial Functions and Their Graphs Sketch the graph of a polynomial by transforming the graph of y = x n, where n is the degree of the polynomial. (Numbers 3 and 5) Match a polynomial with its graph. Be able to justify your answers. (Numbers 11, 13, and 15) Use the factored form of a polynomial to sketch its graph. Show all intercepts of the graph. Make sure the graph exhibits the proper end behavior. (Numbers 21, 25, and 31) Problems 3 and 5: Sketch the graph of the function by transforming an appropriate function of the form y = x n. Indicate all x- and y-intercepts on each graph. P(x) = (x + 2) 3 P(x) = 2x Problems 11, 13, and 15: Match the polynomial function with one of the graphs on page 322 of your textbook. Give reasons for each choice. P(x) = x(x 2 4) R(x) = x 5 + 5x 3 4x T(x) = x 4 + 2x 3 Problems 21 and 25: Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior. P(x) = (x 3)(x + 2)(3x 2) P(x) = 1 12 (x + 2)2 (x 3) 2 Problem 31: Factor the polynomial P(x) = x 3 + x x and use the factored form to find the zeros. Then sketch the graph. 84

4 Section 4.2: Dividing Polynomials Use long division to divide one polynomial by another polynomial. (Numbers 3 and 9) Use synthetic division to divide a polynomial by a polynomial of the form x c. (Number 21) Understand the Remainder Theorem. (Number 35) Understand the Factor Theorem. (Numbers 43 and 53) Problem 3: Let P(x) = x 4 x 3 + 4x + 2 and Q(x) = x (a) Divide P(x) by Q(x). (b) Express P(x) in the form P(x) = D(x) Q(x) + R(x). Problem 9: Find the quotient and remainder using long division for the expression x3 + 6x + 3 x 2 2x + 2. Problem 21: Find the quotient and remainder using synthetic division for the expression x3 8x + 2. x + 3 Problem 35: Use synthetic division and the Remainder Theorem to evaluate P(c) if P(x) = 5x x 3 40x x + 14 and c = 7. Problem 43: Use the Factor Theorem to show that x 1 is a factor of P(x) = x 3 3x 2 + 3x 1. Problem 53: Find a polynomial of degree 3 that has zeros 1, 2, and 3, and in which the coefficient of x 2 is 3. 85

5 Section 4.3: Real Zeros of Polynomials Use the Rational Zeros Theorem (see page 333 of your textbook) to find all rational zeros of a polynomial. (Numbers 3, 13, and 23) Use the Rational Zeros Theorem and the Quadratic Formula if necessary to find all zeros of a polynomial. (Number 43) Problem 3: List all possible rational zeros given by the Rational Zeros Theorem (but don t check to see which actually are zeros). R(x) = 2x 5 + 3x 3 + 4x 2 8. Problems 13 and 23: Find all rational zeros of the polynomial. P(x) = x 3 3x 2 P(x) = x 4 + 6x 3 + 7x 2 6x 8 Problem 43: Find all real zeros of the polynomial P(x) = x 4 7x x 2 3x 9. Use the quadratic formula if necessary, as in Example 3(a) on page 335 of your textbook. 86

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