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Chapter 1: Problem 43

Give an example of two decreasing functions whose product is increasing.

Short Answer

Expert verified
An example of two decreasing functions whose product is increasing are \(f(x) = -x\) and \(g(x) = -x - 1\). Their product is \(h(x) = f(x) \cdot g(x) = x(x + 1)\), which is an increasing function.

Step by step solution

01

Choose the first decreasing function

Let's consider the linear function \(f(x) = -x\), which is a decreasing function.
02

Choose the second decreasing function

Now, let's consider another linear function \(g(x) = -x - 1\), which is also a decreasing function.
03

Compute the product of the two decreasing functions

Now let's compute the product of the two functions: \[h(x) = f(x) \cdot g(x) = (-x) \cdot (-x - 1).\]
04

Simplify the product and check the sign

Simplify the expression for the product function: \[h(x) = (-x)\cdot(-x - 1) = x(x + 1).\] Since \(x\) and \(x + 1\) are both negative for negative values of \(x\), their product, \(h(x)\), will be positive. Additionally, as both \(x\) and \(x + 1\) become less negative as \(x\) increases, the product will also increase. Thus, the functions \(f(x) = -x\) and \(g(x) = -x - 1\) are decreasing functions whose product, \(h(x) = x(x + 1)\), is an increasing function.

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Most popular questions from this chapter

In Exercises \(37-40,\) find functions \(f\) and \(g,\) each simpler than the given function \(h\), such that \(\boldsymbol{h}=\boldsymbol{f} \circ \mathrm{g}\) $$ h(x)=\frac{3}{2+x^{2}} $$

Suppose h is defined by \(h(t)=|t|+1\). What is the range of \(h\) if the domain of \(h\) is the interval [-3,5]\(?\)

For Exercises \(33-40,\) assume that \(f\) is the function defined by $$ f(x)=\left\\{\begin{array}{ll} 2 x+9 & \text { if } x<0 \\ 3 x-10 & \text { if } x \geq 0 \end{array}\right. $$ Evaluate \(f(1)\).

Suppose \(f\) and \(g\) are functions, each of whose domain consists of four numbers, with \(f\) and \(g\) defined by the tables below: $$ \begin{array}{c|c} {x} & {f}({x}) \\ \hline {1} & 4 \\ 2 & 5 \\ 3 & 2 \\ 4 & 3 \end{array} $$ $$ \begin{array}{c|c} x & g(x) \\ \hline 2 & 3 \\ 3 & 2 \\ 4 & 4 \\ 5 & 1 \end{array} $$ Give the table of values for \(g^{-1} \circ f^{-1}\).

Show that the product of two even functions (with the same domain) is an even function.
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