Chapter 1

, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts.. $$ x^{4}+y^{4}=16 $$

Express the solution set of the given inequality in interval notation and sketch its graph. $$ \frac{1}{3 x-2} \leq 4 $$

$$ \text { perform the indicated operations and simplify. } $$ $$ (3 x-9)(2 x+1) $$

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

Solve for \(x .\) Hint: \(\log _{a} b=c \Leftrightarrow a^{c}=b\). $$ \log _{x} 64=4 $$

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. $$ y=3+\sec (x-\pi) $$

Specify whether the given function is even, odd, or neither, and then sketch its graph. $$ g(u)=\frac{u^{3}}{8} $$

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

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

$$ \text { perform the indicated operations and simplify. } $$ $$ (4 x-11)(3 x-7) $$

In Problems 17-22, find the center and radius of the circle with the given equation. 4 x^{2}+16 x+15+4 y^{2}+6 y=0

Solve for \(x .\) Hint: \(\log _{a} b=c \Leftrightarrow a^{c}=b\). $$ 2 \log _{9}\left(\frac{x}{3}\right)=1 $$

Sketch the graph of \(F(t)=\frac{|t|-t}{t}\)

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. $$ y=21+7 \sin (2 x+3) $$

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

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

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

$$ \text { perform the indicated operations and simplify. } $$ $$ \left(3 t^{2}-t+1\right)^{2} $$

In Problems 17-22, find the center and radius of the circle with the given equation. x^{2}+16 x+\frac{105}{16}+4 y^{2}+3 y=0

Solve for \(x .\) Hint: \(\log _{a} b=c \Leftrightarrow a^{c}=b\). $$ \log _{4}\left(\frac{1}{2 x}\right)=3 $$

State whether each of the following is an odd function, an even function, or neither. Prove your statements. (a) The sum of two even functions (b) The sum of two odd functions (c) The product of two even functions (d) The product of two odd functions (e) The product of an even function and an odd function

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. $$ y=3 \cos \left(x-\frac{\pi}{2}\right)-1 $$

Specify whether the given function is even, odd, or neither, and then sketch its graph. $$ \phi(z)=\frac{2 z+1}{z-1} $$

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

Express the solution set of the given inequality in interval notation and sketch its graph. $$ (2 x+3)(3 x-1)(x-2)<0 $$

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

In Problems \(23-28\), find the slope of the line containing the given two points. (1,1) \text { and }(2,2)

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

Let \(F\) be any function whose domain contains \(-x\) whenever it contains \(x\). Prove each of the following. (a) \(F(x)-F(-x)\) is an odd function. (b) \(F(x)+F(-x)\) is an even function. (c) \(F\) can always be expressed as the sum of an odd and an even function.

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. $$ y=\tan \left(2 x-\frac{\pi}{3}\right) $$

Specify whether the given function is even, odd, or neither, and then sketch its graph. $$ f(w)=\sqrt{w-1} $$

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

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

$$ \text { perform the indicated operations and simplify. } $$ $$ \frac{x^{2}-4}{x-2} $$

In Problems \(23-28\), find the slope of the line containing the given two points. (3,5) \text { and }(4,7)

Solve for \(x .\) Hint: \(\log _{a} b=c \Leftrightarrow a^{c}=b\). $$ \log _{5}(x+3)-\log _{5} x=1 $$

Is every polynomial of even degree an even function? Is every polynomial of odd degree an odd function? Explain.

. Which of the following represent the same graph? Check your result analytically using trigonometric identities. (a) \(y=\sin \left(x+\frac{\pi}{2}\right)\) (b) \(y=\cos \left(x+\frac{\pi}{2}\right)\) (c) \(y=-\sin (x+\pi)\) (d) \(y=\cos (x-\pi)\)

Specify whether the given function is even, odd, or neither, and then sketch its graph. $$ h(x)=\sqrt{x^{2}+4} $$

, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts.. $$ 4(x-5)^{2}+9(y+2)^{2}=36 $$

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

$$ \text { perform the indicated operations and simplify. } $$ $$ \frac{x^{2}-x-6}{x-3} $$

In Problems \(23-28\), find the slope of the line containing the given two points. (2,3) \text { and }(-5,-6)

Find each value without using a calculator $$ \cos \left[2 \sin ^{-1}\left(-\frac{2}{3}\right)\right] $$

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

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

, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts.. $$ y=(x-1)(x-2)(x-3) $$

Express the solution set of the given inequality in interval notation and sketch its graph. $$ x^{3}-5 x^{2}-6 x<0 \quad \text { 26. } x^{3}-x^{2}-x+1>0 $$

$$ \text { perform the indicated operations and simplify. } $$ $$ \frac{t^{2}-4 t-21}{t+3} $$

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