Chapter 4

Solve the equation. Check your answers. $$ x^{4}=\frac{1}{81} $$

Solve the rational inequality (a) symbolically and (b) graphically. $$ \frac{x}{x^{2}-1} \geq 0 $$

Complete the following. (a) Find the domain of \(f\) (b) Graph \(f\) in an appropriate viewing rectangle. (c) Find any horizontal or vertical asymptotes. (d) Sketch a graph of \(f\) that includes any asymptotes. $$ f(x)=\frac{4(x-1)}{x^{2}-x-6} $$

Solve the equation. Check your answers. $$ x^{1 / 4}=3 $$

Solve the rational inequality. $$ \frac{(x+1)^{2}}{x-2} \leq 0 $$

Graph \(y=f(x) .\) You may want to use division, factoring, or transformations as an aid. Show all asymptotes and "holes." $$ f(x)=\frac{x^{2}-2 x+1}{x-1} $$

Solve the equation. Check your answers. $$ x^{1 / 3}=\frac{1}{5} $$

Solve the rational inequality. $$ \frac{2 x}{(x-2)^{2}}>0 $$

Graph \(y=f(x) .\) You may want to use division, factoring, or transformations as an aid. Show all asymptotes and "holes." $$ f(x)=\frac{4 x^{2}+4 x+1}{2 x+1} $$

Solve the equation. Check your answers. $$ x^{2 / 5}=4 $$

Solve the rational inequality. $$ \frac{3-2 x}{1+x}<0 $$

Graph \(y=f(x) .\) You may want to use division, factoring, or transformations as an aid. Show all asymptotes and "holes." $$ f(x)=\frac{x+2}{x+1} $$

Solve the equation. Check your answers. $$ x^{2 / 3}=16 $$

Solve the rational inequality. $$ \frac{x+1}{4-2 x} \geq 1 $$

Graph \(y=f(x) .\) You may want to use division, factoring, or transformations as an aid. Show all asymptotes and "holes." $$ f(x)=\frac{2 x+3}{x+1} $$

Solve the equation. Check your answers. $$ 2\left(x^{1 / 5}-2\right)=0 $$

Solve the rational inequality. $$ \frac{(x+1)(x-2)}{(x+3)}<0 $$

Graph \(y=f(x) .\) You may want to use division, factoring, or transformations as an aid. Show all asymptotes and "holes." $$ f(x)=\frac{2 x^{2}-3 x-2}{x^{2}-4 x+4} $$

Solve the equation. Check your answers. $$ x^{1 / 2}+x^{1 / 2}=8 $$

Solve the rational inequality. $$ \frac{x(x-3)}{x+2} \geq 0 $$

Graph \(y=f(x) .\) You may want to use division, factoring, or transformations as an aid. Show all asymptotes and "holes." $$ f(x)=\frac{x^{2}-x-2}{x^{2}-2 x-3} $$

Solve the equation. Check your answers. $$ 4 x^{3 / 2}+5=21 $$

Solve the rational inequality. $$ \frac{2 x-5}{x^{2}-1} \geq 0 $$

Graph \(y=f(x) .\) You may want to use division, factoring, or transformations as an aid. Show all asymptotes and "holes." $$ f(x)=\frac{2 x^{2}+9 x+9}{2 x^{2}+7 x+6} $$

Solve the equation. Check your answers. $$ 2 x^{1 / 3}-5=1 $$

Solve the rational inequality. $$ \frac{5-x}{x^{2}-x-2}<0 $$

Graph \(y=f(x) .\) You may want to use division, factoring, or transformations as an aid. Show all asymptotes and "holes." $$ f(x)=\frac{x^{2}-4}{x^{2}-x-6} $$

Solve the equation. Check your answers. $$ n^{-2}+3 n^{-1}+2=0 $$

Solve the rational inequality. $$ \frac{1}{x-3} \leq \frac{5}{x-3} $$

Graph \(y=f(x) .\) You may want to use division, factoring, or transformations as an aid. Show all asymptotes and "holes." $$ f(x)=\frac{-2 x^{2}+11 x-14}{x^{2}-5 x+6} $$

Solve the equation. Check your answers. $$ 2 n^{-2}-n^{-1}=3 $$

Solve the rational inequality. $$ \frac{3}{2-x}>\frac{x}{2+x} $$

Graph \(y=f(x) .\) You may want to use division, factoring, or transformations as an aid. Show all asymptotes and "holes." $$ f(x)=\frac{2 x^{2}-3 x-14}{x^{2}-2 x-8} $$

The table is a complete representation of \(f .\) Decide if \(f\) is even, odd, or neither. $$\begin{array}{rrrrrrr}x & -5 & -3 & -1 & 1 & 2 & 3 \\ f(x) & -4 & -2 & 1 & 1 & -2 & -4\end{array}$$

Solve the rational inequality. $$ 2-\frac{5}{x}+\frac{2}{x^{2}} \geq 0 $$

Complete the table if \(f\) is an even function. $$\begin{array}{rrrrrrr}x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ f(x) & 21 & & -25 & & & -12 & \end{array}$$

Complete the following. (a) Find any slant or vertical asymptotes. (b) Graph \(y=f(x) .\) Show all asymptotes. $$ f(x)=\frac{x^{2}+1}{x+1} $$

Solve the rational inequality. $$ \frac{1}{x-1}+\frac{1}{x+1}>\frac{3}{4} $$

Complete the table if \(f\) is an odd function. $$\begin{array}{rrrrrrr}x & -5 & -3 & -2 & 0 & 2 & 3 & 5 \\ f(x) & 13 & & -5 & & & -1 & \end{array}$$

Complete the following. (a) Find any slant or vertical asymptotes. (b) Graph \(y=f(x) .\) Show all asymptotes. $$ f(x)=\frac{2 x^{2}-5 x-2}{x-2} $$

Solve the rational inequality. $$ \frac{1}{x} \leq \frac{2}{x+2} $$

If the points \((-5,-6)\) and \((-3,4)\) lie on the graph of an odd function \(f,\) then what \(\operatorname{do} f(5)\) and \(f(3)\) equal?

Complete the following. (a) Find any slant or vertical asymptotes. (b) Graph \(y=f(x) .\) Show all asymptotes. $$ f(x)=\frac{0.5 x^{2}-2 x+2}{x+2} $$

Solve the rational inequality. $$ \frac{1}{x+1}<\frac{1}{x}+1 $$

If the point \((1-a, b+1)\) lies on the graph of an even function \(f,\) then what \(\operatorname{does} f(a-1)\) equal?

Complete the following. (a) Find any slant or vertical asymptotes. (b) Graph \(y=f(x) .\) Show all asymptotes. $$ f(x)=\frac{0.5 x^{2}-5}{x-3} $$

Suppose the average number of vehicles arriving at the main gate of an amusement park is equal to 10 per minute, while the average number of vehicles being admitted through the gate per minute is equal to \(x\). Then the average waiting time in minutes for each vehicle at the gate can be computed by \(f(x)=\frac{x-5}{x^{2}-10 x},\) where \(x>10 .\) (Source: E.Mannering.) (a) Estimate the admittance rate \(x\) that results in an average wait of 15 seconds. (b) If one attendant can serve 5 vehicles per minute, how many attendants are needed to keep the average wait to 15 seconds or less?

Complete the following. (a) Find any slant or vertical asymptotes. (b) Graph \(y=f(x) .\) Show all asymptotes. $$ f(x)=\frac{x^{2}+2 x+1}{x-1} $$

Complete the following. (a) Find any slant or vertical asymptotes. (b) Graph \(y=f(x) .\) Show all asymptotes. $$ f(x)=\frac{2 x^{2}+3 x+1}{x-2} $$

Find possible dimensions for a box with a volume of 196 cubic inches, a surface area of 280 square inches, and a length that is twice the width.