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# College Algebra Study Guide

E.8

27. Show that each function is the inverse of the other: f(x) = 4x - 7 and .
Answer: Yes, the functions are inverse of each other. f(g(x)) = x and g(f(x)) = x.

28. Find the inverse of the function .
Answer: f -1(x) = x2 + 1 for x ≥ 0

29. Does the graph below represent a function that has inverse function?

Answer: No, notice that horizontal lines can be drawn and intersect the graph more than once.

30. The formula is used to convert from x degrees Celsius to y degrees Fahrenheit.
Find the formula to convert from y degrees Fahrenheit to x degrees Celsius. Show that this formula
is the inverse function of f(x).
Answer: is the inverse, then , therefore the
formula is the inverse of f(x).

E.9

31. Graph the polynomial function: f(x) = 2x3-x2-13x-6. Indicate the graph's end behavior, the
x-intercepts, state whether the graph crosses the x-axis or touches the x-axis, indicate the y-intercepts.
If necessary, find a few additional points and graph the function.
Answer: The graph falls to the left and rises to the right, x = -2, , 3, y = -6. Crosses the
x-axis at every zero since each zero has multiplicity 1.

32. Graph the rational function . Indicate all x-intercepts, y-intercepts, horizontal asymp-
tote, vertical asymptote(s). If necessary, find a few additional points and graph the function.
Answer: x = 0, y = 0, vertical asymptote at x = 2, horizontal asymptote at y = 3.

33. Sketch the graph of the exponential function :
Answer: Horizontal asymptote located at y = -3

34. Graph the following Piecewise function and state the domain, range and intervals where the func-
tion is increasing, decreasing and constant.

Answer: The domain of this function is the set of all Real Numbers, the range is the interval
[1,∞). The interval of increasing is (1,∞), the interval of decreasing is (-∞,-1) and the interval
where the function is constant is (-1, 1). See the graph below.

E.10

35. Apply properties of Logarithms to simplify each expression.

36. Expand each expression by writing in terms of sum or difference of logarithms.

37. Write the expression as a single Logarithm.

38. Write the expression as a single Logarithm.

E.11

39. Suppose that y is such that . Evaluate

40. Solve for all the values of x that satisfy the equation: .

41. Solve the equation by making an appropriate substitution, .

42. Solve the radical equation. Check the proposed solutions. .

43. Find the rational zeros of f. List any irrational zero correct to two decimal places.

f(x) = x4 + 5x3 - 3x2 - 35x - 28.

Answer: Rational zeros: , Irrational zeros:

44. Solve the exponential equation. Round your answer to four decimal places.