Chapter 1

Solve the inequality. $$ \frac{2 x\left(x^{2}-3\right)}{\left(x^{2}+1\right)^{3}} \geq 0 $$

Sketch the graph of the equation. In each case determine whether the graph is that of a function. $$ x^{2}+y^{2}=1 $$

Write \(h\) as the composite \(g \circ f\) of two functions \(f\) and \(g\) (neither of which is equal to \(h\) ). $$ h(x)=\sqrt{[x]-1} $$

$$ \text { Solve } \ln x+\ln (3 x-1)=0 \text { for } x \text { . } $$

For a function \(f\), if there is a smallest positive number \(a\) such that \(f(x+a)=f(x)\) for all \(x\) in the domain of \(f\), then \(a\) is called the period of \(f .\) Plot cach of the following functions, and from the graph guess the period. Then prove that your guess is correct. a. \(\tan x\) b. \(\sin 3 x\) c. \(|\sin x|\) d. \(\cos \left(-2 x+\frac{\pi}{2}\right)\)

Decide which pairs of lines are parallel, which are perpendicular, and which are neither. For any pair that is not parallel, find the point of intersection. \(3 y+6 x=1\) and \(y-3=-2 x\)

Sketch the graph of the given equation with the help of a suitable translation. Show both the \(x\) and \(y\) axes and the \(X\) and \(Y\) axes. $$ y+4=\frac{1}{x+2} $$

Solve the inequality. $$ \frac{t^{2}+t-2}{\left(t^{2}-1\right)^{3}} \geq 0 $$

Find the domain of the function. $$ f(x)=\sqrt{1-\sqrt{9-x^{2}}} $$

Sketch the graph of the equation. In each case determine whether the graph is that of a function. $$ x^{2}+y^{2}=9 $$

Find \(g\) if \(f(x)=|x|\) and \((f+g)(x)=|x|-|x-2|\).

Let \(f(x)=\ln \left(x+\sqrt{x^{2}-9}\right)+\ln \left(x-\sqrt{x^{2}-9}\right)\) for \(x \geq\) 3\. Show that \(f\) is a constant function, and find the constant.

Decide which pairs of lines are parallel, which are perpendicular, and which are neither. For any pair that is not parallel, find the point of intersection. \(x-2 y=8\) and \(2 x-y=-8\)

Solve the inequality. $$ \frac{t^{2}-2 t-3}{t^{2}-8 t+15}>0 $$

Find the domain of the function. $$ f(x)=\sqrt{4-\sqrt{1+9 x^{2}}} $$

Sketch the graph of the equation. In each case determine whether the graph is that of a function. $$ x^{2}+y^{2}=9 \text { for } x \geq 0 $$

Find \(g\) if \(f(x)=\left(x^{2}-4\right) /(x+3)\) and \((f g)(x)=1\), for \(x \neq 2\), \(-2\), and \(-3 .\)

Show that \(\ln \left(x+\sqrt{x^{2}-1}\right)=-\ln \left(x-\sqrt{x^{2}-1}\right)\) for \(x \geq 1\).

Find an equation of the line that is parallel to the given line \(l\) and passes through the given point \(P\). \(l: y=3 x-1 ; P=(2,-1)\)

Sketch the graph of the given equation with the help of a suitable translation. Show both the \(x\) and \(y\) axes and the \(X\) and \(Y\) axes. $$ x-2=|y-2| $$

Solve the inequality. $$ \frac{2-x}{\sqrt{9-6 x}}>0 $$

Determine the range of the function. $$ f(x)=-1 $$

Sketch the graph of the equation. In each case determine whether the graph is that of a function. $$ x^{2}+y^{2}=4 \text { for } y \leq 0 $$

Suppose \(f\) is defined on \([0,4]\) and \(g(x)=f(x+3)\). What is the domain of \(g\) ?

Let \(a\) and \(b\) be distinct positive numbers different from \(1 .\) Show that \(\log _{a} x \neq \log _{b} x\) for \(x \neq 1\).

Let \(r\) be any rational number and let \(f(x)=\sin x+\sin r x\). Show that \(f\) is periodic. (Hint: Let \(r=m / n\), where \(m\) and \(n\) are integers.)

Find an equation of the line that is parallel to the given line \(l\) and passes through the given point \(P\). \(l: y=-\frac{1}{2} x+4 ; P=(-1,0)\)

Sketch the graph of the given equation with the help of a suitable translation. Show both the \(x\) and \(y\) axes and the \(X\) and \(Y\) axes. $$ y=\sqrt{x+3} $$

Solve the inequality. $$ \frac{2 x^{2}-1}{\left(1-x^{2}\right)^{1 / 2}}<0 $$

Determine the range of the function. $$ f(x)=3 x-2 $$

Sketch the graph of the equation. In each case determine whether the graph is that of a function. $$ x^{2}+y^{2}=0 $$

Suppose \(f\) is defined on \([a, b]\) and \(g(x)=f(x+c)\) for a fixed \(c\). What is the domain of \(g\) ?

For each of the given values of \(a\), calculate the iterates $$ a, \cos a, \cos (\cos a), \ldots $$ until the first three displayed digits do not change. From your results, make a conjecture about the behavior of the iterates for any real number \(a\). a. \(a=1\) b. \(a=12\) c. \(a=100\) d. \(a=-5\)

Find an equation of the line that is parallel to the given line \(l\) and passes through the given point \(P\). \(l: x+y=1 ; P=(0,0)\)

Determine which of the following functions are even, which are odd, and which are neither. a. \(f(x)=-x\) b. \(f(x)=5 x^{2}-3\) c. \(f(x)=x^{3}+1\) d. \(f(x)=(x-2)^{2}\) e. \(f(x)=\left(x^{2}+3\right)^{3}\) f. \(y=x\left(x^{2}+1\right)^{2}\) g. \(y=\frac{x}{x^{2}+4}\) h. \(y=|x|\) i. \(y=\frac{|x|}{x}\)

Solve the inequality. \(\frac{1}{x+1}>\frac{3}{2} \quad\) (Hint: Write the inequality as \(1 /(x+1)-\) \(3 / 2>0 .\) Then rewrite the left side as a single fraction.)

Determine the range of the function. $$ f(x)=3 x-2 \text { for } x<4 $$

Let \(a>0 .\) Using the Change of Base Formula, show that \(\log _{1 / a} x=-\log _{a} x\) for \(x>0\)

Through how many complete revolutions does a bicycle wheel with radius 1 foot turn when the bicycle travels 1 mile?

Find an equation of the line that is parallel to the given line \(l\) and passes through the given point \(P\). \(l: 3 y+2 x=5 ; P=(-1,-3)\)

Let \(f(x)=a x^{2}+b x+c\), where \(a \neq 0\). Show that any real zeros of \(f\) are given by (1). (Hint: Prove that \(a x^{2}+b x+\) \(c=0\) if and only if $$ \left(x+\frac{b}{2 a}\right)^{2}=\frac{b^{2}-4 a c}{4 a^{2}} $$ Then solve for \(x\). Note that such a (real) zero exists only if \(b^{2}-4 a c \geq 0 .\) There are two zeros if \(b^{2}-4 a c>0\), whereas there is only one zero if \(b^{2}-4 a c=0 .\) )

Solve the inequality. $$ \begin{aligned} &\frac{1}{3-x}<-2 \quad \text { 41. } \frac{x+1}{x-1} \leq \frac{1}{2}\\\ &\text { 42. } \frac{2-5 x}{3-4 x} \geq-2 \end{aligned} $$

Determine the range of the function. $$ f(x)=\sqrt{1-x^{2}} $$

Sketch the graph of the equation. In each case determine whether the graph is that of a function. $$ y=x^{2} \text { for } x \leq 0 $$

Let \(f(x)=\ln (4 x)-\ln x^{3}+\ln x^{2} .\) Plot \(f\) on a graphics calculator, and use properties of logarithms to explain the appearance of the graph.

Solve the inequality. $$ \frac{x+1}{x-1} \leq \frac{1}{2} $$

Find an equation of the line that is parallel to the given line \(l\) and passes through the given point \(P\). \(l:-3 y+2 x=8 ; P=(2,1)\)

Determine the range of the function. $$ f(x)=\frac{1}{x-1} $$

Sketch the graph of the equation. In each case determine whether the graph is that of a function. $$ x^{2}=y^{2} $$

Let \(f(x)=e^{x}\) and \(g(x)=c \ln x .\) Use a graphics calculator to determine an approximate value of \(c\) such that the graphs of \(f\) and \(g\) touch, but do not cross, each other.