Chapter 1

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) \(H(y)=-\sqrt{625-y^{4}}\)

, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts.. $$ x^{2}-y^{2}=4 $$

Find the equation of the circle satisfying the given conditions. Center \((2,-1)\), goes through \((5,3)\)

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. $$ 1-\frac{1}{1+\frac{1}{2}} $$

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

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) \(y=\sec t\)

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) \(F(t)=t^{2 / 3}-4\)

, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts.. $$ x^{2}+(y-1)^{2}=9 $$

Find the equation of the circle satisfying the given conditions. Center \((4,3)\), goes through \((6,2)\)

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.$$ 2+\frac{3}{1+\frac{5}{2}} $$

Use a calculator to approximate each value. $$ \sec ^{-1}(-2.222) $$

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) \(y=\cos \left(t+\frac{\pi}{3}\right)\)

Specify whether the given function is even, odd, or neither, and then sketch its graph. $$ f(x)=-4 $$

, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts.. $$ 4(x-1)^{2}+y^{2}=36 $$

Find the equation of the circle satisfying the given conditions. Diameter \(A B\), where \(A=(1,3)\) and \(B=(3,7)\)

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. $$ (\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3}) $$

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

Use a calculator to approximate each value. $$ \tan ^{-1}(-60.11) $$

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. $$ y=3 \cos \frac{x}{2} $$

Specify whether the given function is even, odd, or neither, and then sketch its graph. $$ f(x)=3 x $$

, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts.. $$ x^{2}-4 x+3 y^{2}=-2 $$

Find the equation of the circle satisfying the given conditions. Center \((3,4)\) and tangent to \(x\) -axis

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.$$ (\sqrt{5}-\sqrt{3})^{2} $$

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

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

Use a calculator to approximate each value. $$ \cos \left(\sin \left(\tan ^{-1} 2.001\right)\right) $$

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. $$ y=2 \sin 2 x $$

Specify whether the given function is even, odd, or neither, and then sketch its graph. $$ F(x)=2 x+1 $$

, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts.. $$ x^{2}+9(y+2)^{2}=36 $$

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

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

$$ \text {17-28, perform the indicated operations and simplify. } $$ $$ (3 x-4)(x+1) $$

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

Solve for \(x .\) Hint: \(\log _{a} b=c \Leftrightarrow a^{c}=b\). $$ \log _{5} x=2 $$

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

Use a calculator to approximate each value. $$ \sin ^{2}(\ln (\cos 0.5555)) $$

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. $$ y=\tan x $$

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

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

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

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

In Problems 17-22, find the center and radius of the circle with the given equation. x^{2}+y^{2}-12 x+35=0

Solve for \(x .\) Hint: \(\log _{a} b=c \Leftrightarrow a^{c}=b\). $$ \log _{4} x=\frac{3}{2} $$

Sketch the graphs of \(f(x)=(x-3) / 2\) and \(g(x)=\sqrt{x}\) using the same coordinate axes. Then sketch \(f+g\) by adding \(y\) -coordinates.

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. $$ y=2+\frac{1}{6} \cot 2 x $$

Specify whether the given function is even, odd, or neither, and then sketch its graph. $$ g(x)=3 x^{2}+2 x-1 $$