Chapter 1

In Problems \(11-16\), find the inverse of the given function \(f\) and verify that \(f\left(f^{-1}(x)\right)=x\) for all \(x\) in the domain of \(f^{-1}\), and \(f^{-1}(f(x))=x\) for all \(x\) in the domain of \(f\). $$ f(x)=\frac{10^{x}}{1+10^{x}} $$

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. $$ x^{2}-y^{2}=4 $$

Write \(p(x)=1 / \sqrt{x^{2}+1}\) as a composite of three functions in two different ways.

Find the natural domain for each of the following. (a) \(F(z)=\sqrt{2 z+3}\) (b) \(g(v)=1 /(4 v-1)\) (c) \(\psi(x)=\sqrt{x^{2}-9}\) (d) \(H(y)=-\sqrt{625-y^{4}}\)

In Problems 11-18, use a calculator to approximate each value. $$ \cos (\operatorname{arcsec} 3.212) $$

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

Simplify as much as possible. Be sure to remove all parentheses and reduce all fractions. \(1-\frac{1}{1+\frac{1}{2}}\)

Sketch the graphs of the following on \([-\pi, 2 \pi]\). (a) \(y=\sin 2 x\) (b) \(y=2 \sin t\) (c) \(y=\cos \left(x-\frac{\pi}{4}\right)\) (d) \(y=\sec t\)

In Problems \(11-16\), find the equation of the circle satisfying the given conditions. Center \((4,3)\), goes through \((6,2)\)

In Problems \(11-16\), find the inverse of the given function \(f\) and verify that \(f\left(f^{-1}(x)\right)=x\) for all \(x\) in the domain of \(f^{-1}\), and \(f^{-1}(f(x))=x\) for all \(x\) in the domain of \(f\). $$ f(x)=\frac{2^{x}}{4+2^{x}} $$

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. $$ x^{2}+(y-1)^{2}=9 $$

Write \(p(x)=1 / \sqrt{x^{2}+1}\) as a composite of four functions.

Find the natural domain in each case. (a) \(f(x)=\frac{4-x^{2}}{x^{2}-x-6}\) (b) \(G(y)=\sqrt{(y+1)^{-1}}\) (c) \(\phi(u)=|2 u+3|\) (d) \(F(t)=t^{2 / 3}-4\)

In Problems 11-18, use a calculator to approximate each value. \(\sec (\arccos 0.5111)\)

Express the solution set of the given inequality in interval notation and sketch its graph. $$ 4 x^{2}-5 x-6<0 $$

Simplify as much as possible. Be sure to remove all parentheses and reduce all fractions. \(2+\frac{3}{1+\frac{5}{2}}\)

Sketch the graphs of the following on \([-\pi, 2 \pi]\). (a) \(y=\csc t\) (b) \(y=2 \cos t\) (c) \(y=\cos 3 t\) (d) \(y=\cos \left(t+\frac{\pi}{3}\right)\)

In Problems \(11-16\), find the equation of the circle satisfying the given conditions. Diameter \(A B\), where \(A=(1,3)\) and \(B=(3,7)\)

In Problems \(11-16\), find the inverse of the given function \(f\) and verify that \(f\left(f^{-1}(x)\right)=x\) for all \(x\) in the domain of \(f^{-1}\), and \(f^{-1}(f(x))=x\) for all \(x\) in the domain of \(f\). $$ f(x)=\log _{10}(3 x+2) $$

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. $$ 4(x-1)^{2}+y^{2}=36 $$

In Problems 15-30, specify whether the given function is even, odd, or neither, and then sketch its graph. \(f(x)=-4\)

In Problems 11-18, use a calculator to approximate each value. \(\sec ^{-1}(-2.222)\)

Express the solution set of the given inequality in interval notation and sketch its graph. $$ \frac{x+4}{x-3} \leq 0 $$

Simplify as much as possible. Be sure to remove all parentheses and reduce all fractions. \((\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})\)

Determine the period, amplitude, and shifts (both horizontal and vertical) and draw a graph over the interval \(-5 \leq x \leq 5\) for the functions listed in Problems 16-23. $$ y=3 \cos \frac{x}{2} $$

In Problems \(11-16\), find the equation of the circle satisfying the given conditions. Center \((3,4)\) and tangent to \(x\)-axis

In Problems \(11-16\), find the inverse of the given function \(f\) and verify that \(f\left(f^{-1}(x)\right)=x\) for all \(x\) in the domain of \(f^{-1}\), and \(f^{-1}(f(x))=x\) for all \(x\) in the domain of \(f\). $$ f(x)=\log _{2}\left(\frac{x+1}{2 x}\right) $$

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. $$ x^{2}-4 x+3 y^{2}=-2 $$

Sketch the graph of \(g(x)=|x+3|-4\) by first sketching \(h(x)=|x|\) and then translating.

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

In Problems 11-18, use a calculator to approximate each value. \(\tan ^{-1}(-60.11)\)

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

Simplify as much as possible. Be sure to remove all parentheses and reduce all fractions. \((\sqrt{5}-\sqrt{3})^{2}\)

Determine the period, amplitude, and shifts (both horizontal and vertical) and draw a graph over the interval \(-5 \leq x \leq 5\) for the functions listed in Problems 16-23. $$ y=2 \sin 2 x $$

In Problems 17-22, find the center and radius of the circle with the given equation. \(x^{2}+2 x+10+y^{2}-6 y-10=0\)

$$ \text { In Problems 17-24, solve for } x . \text { Hint: } \log _{a} b=c \Leftrightarrow a^{c}=b \text {. } $$ $$ \log _{2} 8=x $$

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. $$ x^{2}+9(y+2)^{2}=36 $$

Sketch the graph of \(f(x)=(x-2)^{2}-4\) using translations.

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

In Problems 11-18, use a calculator to approximate each value. \(\cos \left(\sin \left(\tan ^{-1} 2.001\right)\right)\)

Express the solution set of the given inequality in interval notation and sketch its graph. $$ \frac{2}{x}<5 $$

Perform the indicated operations and simplify. \((3 x-4)(x+1)\)

In Problems 17-22, find the center and radius of the circle with the given equation. \(x^{2}+y^{2}-6 y=16\)

$$ \text { In Problems 17-24, solve for } x . \text { Hint: } \log _{a} b=c \Leftrightarrow a^{c}=b \text {. } $$ $$ \log _{5} x=2 $$

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. $$ x^{4}+y^{4}=1 $$

Sketch the graph of \(g(x)=(x+1)^{3}-3\) using translations.

In Problems 15-30, specify whether the given function is even, odd, or neither, and then sketch its graph. \(F(x)=3 x-\sqrt{2}\)

In Problems 11-18, use a calculator to approximate each value. \(\sin ^{2}(\ln (\cos 0.5555))\)

Express the solution set of the given inequality in interval notation and sketch its graph. $$ \frac{7}{4 x} \leq 7 $$

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