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Chapter 0: Problem 72

Conjecture Use examples to hypothesize whether the product of an odd function and an even function is even or odd. Then prove your hypothesis.

Short Answer

Expert verified
The product of an even function and an odd function is an odd function.

Step by step solution

01

Hypothesize using examples

Firstly, let's consider two functions: f(x) = x^2 which is an even function and g(x) = x^3 which is an odd function. The product of f(x) and g(x) is h(x) = x^5, which is an odd function. This example suggests that the product of an even function and an odd function might potentially be an odd function.
02

Prove the hypothesis

Assume we have an arbitrary even function f(x) and odd function g(x). The product function h(x) = f(x) * g(x) is to be determined if it's odd. Since f(-x) = f(x) for the even function and g(-x) = -g(x) for the odd function, substituting -x into h(x): h(-x) = f(-x) * g(-x) implements to h(-x) = f(x) * -g(x) equals -f(x) * g(x). Thus -1 multiplies h(x) which implies h(-x) = -h(x). So it's confirmed that h(x) is an odd function.

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Most popular questions from this chapter

In Exercises 75-78, use the functions given by \(f(x)=x+4\) and \(g(x)=2 x-5\) to find the specified function. $$ f^{-1} \circ g^{-1} $$

(a) use the position equation \(s=-16 t^{2}+v_{0} t+s_{0}\) to write a function that represents the situation, (b) use a graphing utility to graph the function, (c) find the average rate of change of the function from \(t_{1}\) to \(t_{2}\), (d) interpret your answer to part (c) in the context of the problem, (e) find the equation of the secant line through \(t_{1}\) and \(t_{2}\), and (f) graph the secant line in the same viewing window as your position function. An object is thrown upward from ground level at a velocity of 96 feet per second. $$ t_{1}=2, t_{2}=5 $$

In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=x^{3}+1 $$

You need a total of 50 pounds of two types of ground beef costing \(\$ 1.25\) and \(\$ 1.60\) per pound, respectively. A model for the total cost \(y\) of the two types of beef is $$ y=1.25 x+1.60(50-x) $$ where \(x\) is the number of pounds of the less expensive ground beef. (a) Find the inverse function of the cost function. What does each variable represent in the inverse function? (b) Use the context of the problem to determine the domain of the inverse function. (c) Determine the number of pounds of the less expensive ground beef purchased when the total cost is \(\$ 73\).

In Exercises 55-68, determine whether the function has an inverse function. If it does, find the inverse function. $$ f(x)=\frac{3 x+4}{5} $$
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