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Chapter 1: Problem 123

If \(f\) is an even function, determine whether \(g\) is even, odd, or neither. Explain. (a) \(g(x)=-f(x)\) (b) \(g(x)=f(-x)\) (c) \(g(x)=f(x)-2\) (d) \(g(x)=f(x-2)\)

Short Answer

Expert verified
(a) g is odd, (b) g is even, (c) g is even, and (d) g cannot be determined to be even, odd, or neither without further information about f.

Step by step solution

01

Determine Type of \(g(x) = -f(x)\)

Substitute \(-x\) in place of \(x\) in \(g(x)\): \(g(-x) = -f(-x)\). Since \(f\) is an even function, this can be written as \(g(-x) = -f(x)\). Consequently, if for all \(x\), \(g(-x) = -g(x)\), \(g\) is an odd function.
02

Determine Type of \(g(x) = f(-x)\)

Substitute \(-x\) in place of \(x\) in \(g(x)\): \(g(-x) = f(x)\). This is the definition of an even function. Therefore, if for all \(x\), \(g(-x) = g(x)\), \(g\) is an even function.
03

Determine Type of \(g(x) = f(x) - 2\)

Substitute \(-x\) in place of \(x\) in \(g(x)\): \(g(-x) = f(-x) - 2\). Because \(f\) is an even function, this can be simplified to \(g(-x) = f(x) - 2 = g(x)\). So, if for all \(x\), \(g(-x) = g(x)\), \(g\) is an even function.
04

Determine Type of \(g(x) = f(x - 2)\)

Substitute \(-x\) for \(x\) in \(g(x)\): \(g(-x) = f(-x - 2)\). This cannot be simplified using the properties of even functions. Thus, without more specific information about \(f\), it cannot be determined if \(g\) is even, odd, or neither.

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Most popular questions from this chapter

(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 \(f^{-1}\). $$ f(x)=\sqrt[3]{x-1} $$

The direct variation model \(y=k x^{n}\) can be described as " \(y\) varies directly as the \(n\) th power of \(x\)," or "y is _____ _____to the \(n\) th power of \(x.\) "

Restrict the domain of \(f(x)=x^{2}+1\) to \(x \geq 0 .\) Use a graphing utility to graph the function. Does the restricted function have an inverse function? Explain.

Use the functions given by \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the indicated value or function. $$ \left(g^{-1} \circ f^{-1}\right)(-3) $$

Determine whether the variation model is of the form \(y=k x\) or \(y=k / x,\) and find \(k .\) Then write \(a\) model that relates \(y\) and \(x\). $$ \begin{array}{|c|c|c|c|c|c|} \hline x & 5 & 10 & 15 & 20 & 25 \\ \hline y & -3.5 & -7 & -10.5 & -14 & -17.5 \\ \hline \end{array} $$
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