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Chapter 1: Problem 67

Suppose \(f\) is an even function and \(g\) is an odd function. Show that \(f \circ g\) is an even function.

Short Answer

Expert verified
The composition of an even function \(f\) and an odd function \(g\) is even, as shown by the equation \(f(g(-x)) = f(-g(x)) = f(g(x))\).

Step by step solution

01

Write the definitions of even and odd functions

Even function: \(f(-x) = f(x)\) for all \(x\). Odd function: \(g(-x) = -g(x)\) for all \(x\).
02

Write the composition \(f(g(x))\)

The composition of functions \(f\) and \(g\) is given by: \(f(g(x))\).
03

Replace \(x\) with \(-x\) in the composition

We substitute \(x\) with \(-x\) in the composition: \(f(g(-x))\).
04

Apply the properties of even and odd functions

We use the property of odd functions from Step 1: \(g(-x) = -g(x)\). So, we have: \(f(-g(x))\). Now, we apply the property of even functions from Step 1: \(f(-x) = f(x)\). So, we get: \(f(g(-x)) = f(g(x))\).
05

Identify the final result

We have shown that \(f(g(-x)) = f(g(x))\), which is the property of even functions. Therefore, \(f \circ g\) is an even function.

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Most popular questions from this chapter

Suppose \(f\) and \(g\) are functions, each with domain 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 \(f^{-1} \circ f\).

Suppose \(f\) is a function and a function \(g\) is defined by the given expression. (a) Write \(g\) as the composition of \(f\) and one or two linear functions. (b) Describe how the graph of \(g\) is obtained from the graph of \(f\). \( g(x)=f\left(-\frac{2}{3} x\right)\)

Suppose \(f\) is a function and a function \(g\) is defined by the given expression. (a) Write \(g\) as the composition of \(f\) and one or two linear functions. (b) Describe how the graph of \(g\) is obtained from the graph of \(f\). \( g(x)=2 f(3 x)+4\)

Suppose \(f\) is a function and a function \(g\) is defined by the given expression. (a) Write \(g\) as the composition of \(f\) and one or two linear functions. (b) Describe how the graph of \(g\) is obtained from the graph of \(f\). \( g(x)=-5 f\left(-\frac{4}{3} x\right)-8\)

A temperature \(F\) degrees Fahrenheit corresponds to \(g(F)\) degrees on the Kelvin temperature scale, where $$g(F)=\frac{5}{9} F+255.37$$ (a) Find a formula for \(g^{-1}(K)\). (b) What is the meaning of \(g^{-1}(K) ?\) (c) Evaluate \(g^{-1}(0)\). (This is absolute zero, the lowest possible temperature, because all molecular activity stops at 0 degrees Kelvin.)
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