Chapter 1

In Problems 25-28, find each value without using a calculator (see Example 4). $$ \cos \left[2 \sin ^{-1}\left(-\frac{2}{3}\right)\right] $$

Express the solution set of the given inequality in interval notation and sketch its graph. $$ x^{3}-5 x^{2}-6 x<0 $$

Perform the indicated operations and simplify. \(\frac{t^{2}-4 t-21}{t+3}\)

Which of the following are odd functions? Even functions? Neither? (a) \(\cot t+\sin t\) (b) \(\sin ^{3} t\) (c) \(\sec t\) (d) \(\sqrt{\sin ^{4} t}\) (e) \(\cos (\sin t)\) (f) \(x^{2}+\sin x\)

In Problems 23-28, find the slope of the line containing the given two points. \((2,-4)\) and \((0,-6)\)

In Problems 1-30, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\)-intercepts. $$ y=x^{2}(x-1)(x-2) $$

In Problems 15-30, specify whether the given function is even, odd, or neither, and then sketch its graph. \(F(t)=-|t+3|\)

Express the solution set of the given inequality in interval notation and sketch its graph. $$ x^{3}-x^{2}-x+1>0 $$

Perform the indicated operations and simplify. \(\frac{2 x-2 x^{2}}{x^{3}-2 x^{2}+x}\)

In Problems 23-28, find the slope of the line containing the given two points. \((3,0)\) and \((0,5)\)

In Problems 25-28, the graph of an exponential function of the form \(y=C a^{x}\) is given. Use the graph to determine a and \(C\).

In Problems 1-30, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\)-intercepts. $$ y=x^{2}(x-1)^{2} $$

In Problems 15-30, specify whether the given function is even, odd, or neither, and then sketch its graph. \(g(x)=\left\lceil\frac{x}{2}\right\rceil\)

Tell whether each of the following is true or false. (a) \(-3<-7\) (b) \(-1>-17\) (c) \(-3<-\frac{22}{7}\)

Perform the indicated operations and simplify. \(\frac{12}{x^{2}+2 x}+\frac{4}{x}+\frac{2}{x+2}\)

Find the exact values in Problems 27-31. Hint: Half-angle identities may be helpful. $$ \sin ^{2} \frac{\pi}{6} $$

In Problems 23-28, find the slope of the line containing the given two points. \((-6,0)\) and \((0,6)\)

In Problems 1-30, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\)-intercepts. $$ y=x^{4}(x-1)^{4}(x+1)^{4} $$

In Problems 15-30, specify whether the given function is even, odd, or neither, and then sketch its graph. \(G(x)=[2 x-1]\)

In Problems 25-28, find each value without using a calculator (see Example 4). $$ \cos \left[\cos ^{-1}\left(\frac{4}{5}\right)+\sin ^{-1}\left(\frac{12}{13}\right)\right] $$

Tell whether each of the following is true or false. (a) \(-5>-\sqrt{26}\) (b) \(\frac{6}{7}<\frac{34}{39}\) (c) \(-\frac{5}{7}<-\frac{44}{59}\)

Perform the indicated operations and simplify. \(\frac{2}{6 y-2}+\frac{y}{9 y^{2}-1}\)

Find the exact values in Problems 27-31. Hint: Half-angle identities may be helpful. $$ \sin ^{3} \frac{\pi}{6} $$

In Problems \(29-34\), find an equation for each line. Then write your answer in the form \(A x+B y+C=0\). Through \((2,2)\) with slope \(-1\)

In Problems 1-30, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\)-intercepts. $$ |x|+|y|=1 $$

In Problems 15-30, specify whether the given function is even, odd, or neither, and then sketch its graph. \(g(t)= \begin{cases}1 & \text { if } t \leq 0 \\ t+1 & \text { if } 0

In Problems 29-32, show that each equation is an identity. \(\tan \left(\sin ^{-1} x\right)=\frac{x}{\sqrt{1-x^{2}}}\)

Assume that \(a>0, b>0\). Prove each statement. Hint: Each part requires two proofs: one for \(\Rightarrow\) and one for \(\Leftarrow\). (a) \(a\frac{1}{b}\)