91Ó°ÊÓ

Give an example of a function \(g\) with the property that \(g(x)=g(-x)\) for every real number \(x\)

Give an example of a function \(g\) with the property that \(g(-x)=-g(x)\) for every real number \(x\)

$$\text {find the values of } x \text { for which } f(x)=g(x)$$. $$f(x)=2 x^{2}+4 x-4 ; \quad g(x)=x^{2}+12 x+6$$

$$\text {find the values of } x \text { for which } f(x)=g(x)$$. $$f(x)=2 x^{2}+13 x-14 ; \quad g(x)=8 x-2$$

$$\text {find the values of } x \text { for which } f(x)=g(x)$$. $$f(x)=3 x^{2}-x+5 ; \quad g(x)=x^{2}-2 x+26$$

Use the standard viewing window to graph the function f and the function \(g(x)=|f(x)|\) on the same screen. Exercise 66 may be helpful for interpreting the results. $$f(x)=x+3$$

$$\text {Find the values of } x \text { for which } f(x)=g(x)$$. $$f(x)=2 x^{2}-x+1 ; \quad g(x)=x^{2}-4 x+4$$

Find a function \(f\) (other than the identity function) such that \((f \circ f \circ f)(x)=x\) for every \(x\) in the domain of \(f .\) [Several correct answers are possible.]

If \(f\) is an increasing function, does \(f \circ f\) have to be increasing? Why or why not?

(a) Let \(f\) be a function, and let \(g\) be the function defined by \(g(x)=|f(x)| .\) Use the definition of absolute value (page 9) to explain why the following statement is true: $$g(x)=\left\\{\begin{array}{ll}f(x) & \text { if } f(x) \geq 0 \\\\-f(x) & \text { if } f(x)<0\end{array}\right.$$ (b) Use part (a) and your knowledge of transformations to explain why the graph of \(g\) consists of those parts of the graph of \(f\) that lie above the \(x\) -axis together with the reflection in the \(x\) -axis of those parts of the graph of \(f\) that lie below the \(x\) -axis.