Chapter 1

, simplify as much as possible. Be sure to remove all parentheses and reduce all fractions.$$ \frac{3}{4-7}+\frac{3}{21}-\frac{1}{6} $$

In Problems 7-10, sketch a graph of the given logarithmic function. $$ f(x)=\log _{5} x $$

Calculate \(g(3.141)\) if \(g(u)=\frac{\sqrt{u^{3}+2 u}}{2+u} .\)

find the exact value without using a calculator. $$ \arcsin \left(-\frac{1}{2}\right) $$

. Calculate. (a) \(\frac{56.3 \tan 34.2^{\circ}}{\sin 56.1^{\circ}}\) (b) \(\left(\frac{\sin 35^{\circ}}{\sin 26^{\circ}+\cos 26^{\circ}}\right)^{3}\)

Which of the following determine a function \(f\) with formula \(y=f(x) ?\) For those that do, find \(f(x)\). Hint: Solve for \(y\) in terms of \(x\) and note that the definition of a function requires a single \(y\) for each \(x\). (a) \(x^{2}+y^{2}=1\) (b) \(x y+y+x=1, x \neq-1\) (c) \(x=\sqrt{2 y+1}\) (d) \(x=\frac{y}{y+1}\)

, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts.. $$ 7 x^{2}+3 y=0 $$

Express the solution set of the given inequality in interval notation and sketch its graph. $$ -4<3 x+2<5 $$

, simplify as much as possible. Be sure to remove all parentheses and reduce all fractions.$$ \frac{1}{3}\left[\frac{1}{2}\left(\frac{1}{4}-\frac{1}{3}\right)+\frac{1}{6}\right] $$

Sketch a graph of the given logarithmic function. $$ f(x)=\log _{3} x $$

Calculate \(g(2.03)\) if \(g(x)=\frac{(\sqrt{x}-\sqrt[3]{x})^{4}}{1-x+x^{2}}\)

find the exact value without using a calculator. $$ \tan ^{-1}\left(-\frac{\sqrt{3}}{3}\right) $$

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

Find the point on the \(x\) -axis that is equidistant from \((3,1)\) and \((6,4)\).

Express the solution set of the given inequality in interval notation and sketch its graph. $$ -3<4 x-9<11 $$

, simplify as much as possible. Be sure to remove all parentheses and reduce all fractions. $$ -\frac{1}{3}\left[\frac{2}{5}-\frac{1}{2}\left(\frac{1}{3}-\frac{1}{5}\right)\right] $$

Sketch a graph of the given logarithmic function. $$ f(x)=\log _{2}(x-1) $$

Calculate \(\left[g^{2}(\pi)-g(\pi)\right]^{1 / 3}\) if \(g(v)=|11-7 v|\).

find the exact value without using a calculator. $$ \sin \left(\sin ^{-1} 0.4567\right) $$

Evaluate without using a calculator. (a) \(\tan \frac{\pi}{6}\) (b) \(\sec \pi\) (c) \(\sec \frac{3 \pi}{4}\)

For \(f(x)=2 x^{2}-1\), find and simplify \([f(a+h)-\) \(f(a)] / h\)

, 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 distance between \((-2,3)\) and the midpoint of the segment joining \((-2,-2)\) and \((4,3)\).

Express the solution set of the given inequality in interval notation and sketch its graph. $$ -3<1-6 x \leq 4 $$

, simplify as much as possible. Be sure to remove all parentheses and reduce all fractions.$$ \frac{14}{21}\left(\frac{2}{5-\frac{1}{3}}\right)^{2} $$

Sketch a graph of the given logarithmic function. $$ f(x)=\log _{10}(x+2) $$

Calculate \(\left[g^{3}(\pi)-g(\pi)\right]^{1 / 3}\) if \(g(x)=6 x-11\).

Evaluate without using a calculator. (a) \(\tan \frac{\pi}{3}\) (b) \(\sec \frac{\pi}{3}\) (c) \(\cot \frac{\pi}{3}\) (d) \(\csc \frac{\pi}{4}\) (e) \(\tan \left(-\frac{\pi}{6}\right)\) (f) \(\cos \left(-\frac{\pi}{3}\right)\)

For \(F(t)=4 t^{3}\), find and simplify \([F(a+h)-F(a)] / h .\)

, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts.. $$ 3 x^{2}+4 y^{2}=12 $$

Find the length of the line segment joining the midpoints of the segments \(A B\) and \(C D\), where \(A=(1,3), B=(2,6)\), \(C=(4,7)\), and \(D=(3,4)\).

Express the solution set of the given inequality in interval notation and sketch its graph. $$ 4<5-3 x<7 $$

, simplify as much as possible. Be sure to remove all parentheses and reduce all fractions. $$ \left(\frac{2}{7}-5\right) /\left(1-\frac{1}{7}\right) $$

Use a calculator to approximate each value. \(\sin ^{-1}(0.1113)\)

Verify that the following are identities (see Example 6). (a) \((1+\sin z)(1-\sin z)=\frac{1}{\sec ^{2} z}\) (b) \((\sec t-1)(\sec t+1)=\tan ^{2} t\) (c) \(\sec t-\sin t \tan t=\cos t\) (d) \(\frac{\sec ^{2} t-1}{\sec ^{2} t}=\sin ^{2} t\)

For \(g(u)=3 /(u-2)\), find and simplify \([g(x+h)-\) \(g(x)] / h\)

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

In Problems 11-16, find the equation of the circle satisfying the given conditions. Center \((1,1)\), radius 1

Express the solution set of the given inequality in interval notation and sketch its graph. $$ x^{2}+2 x-12<0 $$

, simplify as much as possible. Be sure to remove all parentheses and reduce all fractions. $$ \frac{\frac{11}{7}-\frac{12}{21}}{\frac{11}{7}+\frac{12}{21}} $$

Find the inverse of the given function fand verify that \(f\left(f^{-1}(x)\right)=x\) for all \(x\) in the domain of \(f^{-1}\), and \(50^{-1}(f(x))=x\) for all \(x\) in the domain off. $$ f(x)=3+10^{x} $$

Find \(f\) and \(g\) so that \(p=f \circ g\). (a) \(p(x)=\frac{2}{\left(x^{2}+x+1\right)^{3}}\) (b) \(p(x)=\frac{1}{x^{3}+3 x}\)

Use a calculator to approximate each value. $$ \arccos (0.6341) $$

For \(G(t)=t /(t+4)\), find and simplify \([G(a+h)-\) \(G(a)] / h .\)

, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts.. $$ 4 x^{2}+3 y^{2}=12 $$

Find the equation of the circle satisfying the given conditions. Center \((-2,3)\), radius 4

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

simplify as much as possible. Be sure to remove all parentheses and reduce all fractions. $$ \frac{\frac{1}{2}-\frac{3}{4}+\frac{7}{8}}{\frac{1}{2}+\frac{3}{4}-\frac{7}{8}} $$

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

Verify the following are identities. (a) \(\frac{\sin u}{\csc u}+\frac{\cos u}{\sec u}=1\) (b) \(\left(1-\cos ^{2} x\right)\left(1+\cot ^{2} x\right)=1\) (c) \(\sin t(\csc t-\sin t)=\cos ^{2} t\) (d) \(\frac{1-\csc ^{2} t}{\csc ^{2} t}=\frac{-1}{\sec ^{2} t}\)