91Ó°ÊÓ

Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other. $$f(x)=x^{3}+1 ; \quad g(x)=(x-1)^{1 / 3}$$

Graph the function in each of the given viewing rectangles, and select the one that produces the most appropriate graph of the function. \(k(x)=\frac{1}{32} x^{4}-x^{2}+2\) (a) \([-1,1]\) by \([-1,1]\) (b) \([-2,2]\) by \([-2,2]\) (c) \([-5,5]\) by \([-5,5]\) (d) \([-10,10]\) by \([-10,10]\)

Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$y=\sqrt[3]{-x}$$

If \(f(x)=m x+b\) is a linear function, then the average rate of change of \(f\) between any two real numbers \(x_{1}\) and \(x_{2}\) is $$\text { average rate of change }=\frac{f\left(x_{2}\right)-f\left(x_{1}\right)}{x_{2}-x_{1}}$$ Calculate this average rate of change to show that it is the same as the slope \(m\)

Use the function to evaluate the indicated expressions and simplify. $$f(x)=3 x-1 ; \quad f(2 x), 2 f(x)$$

Find the functions \(f \circ g, g \circ f, f \circ f,\) and \(g \circ g\) and their domains. $$f(x)=2 x+3, \quad g(x)=4 x-1$$

Use the function to evaluate the indicated expressions and simplify. $$f(x)=x+4 ; \quad f\left(x^{2}\right),(f(x))^{2}$$

Sketch the graph of the piecewise defined function. \(f(x)=\left\\{\begin{array}{ll}0 & \text { if } x<2 \\ 1 & \text { if } x \geq 2\end{array}\right.\)

Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$y=\frac{1}{4} x^{2}$$

If the function \(f\) has the same average rate of change \(c\) between any two points, then for the points \(a\) and \(x\) we have $$c=\frac{f(x)-f(a)}{x-a}$$ Rearrange this expression to show that $$f(x)=c x+(f(a)-c a)$$ and conclude that \(f\) is a linear function.