Chapter 1

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

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

$$ \text { In Problems 17-24, solve for } x . \text { Hint: } \log _{a} b=c \Leftrightarrow a^{c}=b \text {. } $$ $$ \log _{4} x=\frac{3}{2} $$

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.

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. $$ x^{4}+y^{4}=16 $$

In Problems 15-30, specify whether the given function is even, odd, or neither, and then sketch its graph. \(g(x)=3 x^{2}+2 x-1\)

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

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. \(x^{2}+y^{2}-10 x+10 y=0\)

$$ \text { In Problems 17-24, solve for } x . \text { Hint: } \log _{a} b=c \Leftrightarrow a^{c}=b \text {. } $$ $$ \log _{x} 64=4 $$

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. $$ y=x^{3}-x $$

In Problems 15-30, specify whether the given function is even, odd, or neither, and then sketch its graph. \(g(u)=\frac{u^{3}}{8}\)

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

Perform the indicated operations and simplify. \((4 x-11)(3 x-7)\)

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

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

$$ \text { In Problems 17-24, solve for } x . \text { Hint: } \log _{a} b=c \Leftrightarrow a^{c}=b \text {. } $$ $$ 2 \log _{9}\left(\frac{x}{3}\right)=1 $$

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

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

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

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

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

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

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

$$ \text { In Problems 17-24, solve for } x . \text { Hint: } \log _{a} b=c \Leftrightarrow a^{c}=b \text {. } $$ $$ \log _{4}\left(\frac{1}{2 x}\right)=3 $$

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

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

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

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

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

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

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

$$ \text { In Problems 17-24, solve for } x . \text { Hint: } \log _{a} b=c \Leftrightarrow a^{c}=b \text {. } $$ $$ \log _{2}(x+3)-\log _{2} x=2 $$

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.

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. $$ 2 x^{2}-4 x+3 y^{2}+12 y=-2 $$

In Problems 15-30, specify whether the given function is even, odd, or neither, and then sketch its graph. \(f(w)=\sqrt{w-1}\)

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

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

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) \(y=\cos (x-\pi)\) (e) \(y=-\sin (\pi-x)\) (f) \(y=\cos \left(x-\frac{\pi}{2}\right)\) (g) \(y=-\cos (\pi-x)\) (h) \(y=\sin \left(x-\frac{\pi}{2}\right)\)

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

$$ \text { In Problems 17-24, solve for } x . \text { Hint: } \log _{a} b=c \Leftrightarrow a^{c}=b \text {. } $$ $$ \log _{5}(x+3)-\log _{5} x=1 $$

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

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. $$ 4(x-5)^{2}+9(y+2)^{2}=36 $$

In Problems 15-30, specify whether the given function is even, odd, or neither, and then sketch its graph. \(h(x)=\sqrt{x^{2}+4}\)

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

Perform the indicated operations and simplify. \(\frac{x^{2}-x-6}{x-3}\)

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) \(x+\sin x\)

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

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. $$ y=(x-1)(x-2)(x-3) $$

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