91Ó°ÊÓ

Could the following statement ever be the rule of a function? For input \(x,\) the output is the number whose square is \(x\) Why or why not? If there is a function with this rule, what is its domain and range?

Find the approximate intervals on which the function is increasing, those on which it is decreasing, and those on which it is constant. $$f(x)=\frac{1}{x}$$

Fill the blanks in the given table. In each case the values of the functions \(f\) and \(g\) are given by these tables: $$\begin{array}{|c|c|} \hline x & f(x) \\ \hline 1 & 3 \\ \hline 2 & 5 \\\ \hline 3 & 1 \\ \hline 4 & 2 \\ \hline 5 & 3 \\ \hline \end{array}$$ $$\begin{array}{|c|c|} \hline t & g(t) \\ \hline 1 & 5 \\ \hline 2 & 4 \\\ \hline 3 & 4 \\ \hline 4 & 3 \\ \hline 5 & 2 \\ \hline \end{array}$$ $$\begin{array}{|c|c|} \hline t & (f \circ g)(t) \\ \hline 1 & \\ \hline 2 & 2 \\ \hline 3 & \\ \hline 4 & \\ \hline 5 & \\ \hline \end{array}$$

Each given function has an inverse function. Sketch the graph of the inverse function. $$f(x)=\left\\{\begin{array}{ll}x^{2}-1 & \text { if } x \leq 0 \\\\-5 x-1 & \text { if } x>0\end{array}\right.$$

Find the radius \(r\) and height \(h\) of a cylindrical can with a surface area of 60 square inches and the largest possible volume, as follows. (a) Write an equation for the volume \(V\) of the can in terms of \(r\) and \(h\). (b) Write an equation in \(r\) and \(h\) that expresses the fact that the surface area of the can is \(60 .\) [ Hint: Think of cutting the top and bottom off the can; then cut the side of the can lengthwise and roll it out flat; it's now a rectangle. The surface area is the area of the top and bottom plus the area of this rectangle. The length of the rectangle is the same as the circumference of the original can (why?).] (c) Write an equation that expresses \(V\) as a function of \(r\) [Hint: Solve the equation in part (b) for \(h\), and substitute the result in the equation of part (a).] (d) Graph the function in part (c), and find the value of \(r\) that produces the largest possible value of \(V\). What is \(h\) in this case?

Write the given function as the composite of two functions, neither of which is the identity function, as in Examples 6 and 7 . (There may be more than one way to do this.) $$h(x)=\left(7 x^{3}-10 x+17\right)^{7}$$

Write the given function as the composite of two functions, neither of which is the identity function, as in Examples 6 and 7 . (There may be more than one way to do this.) $$ \begin{array}{l} \text { If } f(x)=x+1 \text { and } g(t)=t^{2} \text { , then } \\ \qquad \begin{aligned} (g \circ f)(x) &=g(f(x))=g(x+1)=(x+1)^{2} \\\ &=x^{2}+2 x+1 \end{aligned} \end{array} $$ Find two other functions \(h(x)\) and \(k(t)\) such that \((k \circ h)(x)=x^{2}+2 x+1\)

The integer part function has the set of all real numbers (written as decimals) as its domain. The rule is "For each input number, the output is the part of the number to the left of the decimal point." For instance, the input 37.986 produces the output \(37,\) and the input -1.5 produces the output \(-1 .\) On most calculators, the integer part function is denoted "iPart." On calculators that use "Intg" or "Floor" for the greatest integer function, the integer part function is denoted by "INT." (a) For each nonnegative real number input, explain why both the integer part function and the greatest integer function [Example \(7]\) produce the same output. (b) For which negative numbers do the two functions produce the same output? (c) For which negative numbers do the two functions produce different outputs?

Draw the graph of a function \(f\) that satisfies the following four conditions: (i) domain \(f=[-2,4]\) (ii) range \(f=[-5,6]\) (iii) \(f(-1)=f(3)\) (iv) \(f\left(\frac{1}{2}\right)=0\)

Sketch the graph of the equation. $$|x|+|y|=1$$