Chapter 1

Find the domain of the function. $$ g(x)=\frac{2}{x-1} $$

Write \(h\) as the composite \(g \circ f\) of two functions \(f\) and \(g\) (neither of which is equal to \(h\) ). $$ h(x)=\sqrt{x-3} $$

Sketch the graph of \(f\). $$ f(x)=\ln (e x) $$

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. \(y=3-2 x\) and \(3 x+\frac{3}{2} y-4=0\)

Solve the equation \(4 \cos ^{2} x-4 \sqrt{3} \cos x+3=0\).

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. $$ (x+2)^{2}+(y+4)^{2}=\frac{1}{4} $$

Solve the inequality. $$ \frac{(2 x-3)(4 x+1)}{x-2} \leq 0 $$

Have a computer or graphics calculator plot the graph of the function on the given \(x\) window. Alter the \(y\) window as needed in order to be able to discern the major features of the graph. By reading off the \(y\) coordinate of the lowest point on the graph, approximate the smallest value assumed by the function on the given interval. It may be helpful to adjust the window. $$ f(x)=2 x^{2}-10 x+13 ;[0,5] $$

Find the domain of the function. $$ g(x)=\frac{3 x-1}{x-3} $$

Write \(h\) as the composite \(g \circ f\) of two functions \(f\) and \(g\) (neither of which is equal to \(h\) ). $$ h(x)=\left(1-x^{2}\right)^{3 / 2} $$

Determine at which points the graphs of the given pair of functions intersect. $$ f(x)=e^{x} \text { and } g(x)=e^{1-x} \text { (Hint: Take natural logarithms.) } $$

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. \(x-y=-1\) and \(x=y\)

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

Solve the inequality. $$ 4 x^{3}-6 x^{2} \leq 0 $$

Have a computer or graphics calculator plot the graph of the function on the given \(x\) window. Alter the \(y\) window as needed in order to be able to discern the major features of the graph. By reading off the \(y\) coordinate of the lowest point on the graph, approximate the smallest value assumed by the function on the given interval. It may be helpful to adjust the window. $$ f(x)=x^{7}-x^{5} ;[-1,1] $$

Find the domain of the function. $$ g(w)=\frac{2 w-8}{w^{2}-16} $$

Write \(h\) as the composite \(g \circ f\) of two functions \(f\) and \(g\) (neither of which is equal to \(h\) ). $$ h(x)=\left(3 x^{2}-5 \sqrt{x}\right)^{1 / 3} $$

Determine at which points the graphs of the given pair of functions intersect. $$ f(x)=e^{x} \text { and } g(x)=e^{-x^{2}} $$

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. \(x+y=-1\) and \(x=y\)

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. $$ x^{2}+y^{2}+4 y=-1 $$

Solve the inequality. $$ 3 x^{2}-2 x-1 \geq 0 $$

Have a computer or graphics calculator plot the graph of the function on the given \(x\) window. Alter the \(y\) window as needed in order to be able to discern the major features of the graph. By reading off the \(y\) coordinate of the lowest point on the graph, approximate the smallest value assumed by the function on the given interval. It may be helpful to adjust the window. $$ f(x)=x^{2}(x-5)^{2} ;[-3,7] $$

Find the domain of the function. $$ g(w)=\frac{w-1}{w^{2}-w-6} $$

Write \(h\) as the composite \(g \circ f\) of two functions \(f\) and \(g\) (neither of which is equal to \(h\) ). $$ h(x)=\left(x+\frac{1}{x}\right)^{5 / 2} $$

Determine at which points the graphs of the given pair of functions intersect. $$ f(x)=e^{3 x} \text { and } g(x)=3 e^{x} $$

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. \(2 x+3 y=-1\) and \(2 y+2=3(x-1)\)

Solve the inequality. $$ 8 x-\frac{1}{x^{2}}>0 $$

Have a computer or graphics calculator plot the graph of the function on the given \(x\) window. Alter the \(y\) window as needed in order to be able to discern the major features of the graph. By reading off the \(y\) coordinate of the lowest point on the graph, approximate the smallest value assumed by the function on the given interval. It may be helpful to adjust the window. $$ f(x)=\frac{4 x^{3}-x+1}{x^{4}+1} ;[-3,3] $$

Find the domain of the function. $$ k(x)=\frac{2 x-3}{x^{2}+4} $$

Write \(h\) as the composite \(g \circ f\) of two functions \(f\) and \(g\) (neither of which is equal to \(h\) ). $$ h(x)=\frac{1}{(x+3)^{2}+1} $$

Determine at which points the graphs of the given pair of functions intersect. $$ f(x)=5 e^{-2 x} \text { and } g(x)=3 e^{x} $$

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. \(2 x+3 y=-1\) and \(3 x+2 y=2\)

a. Using \((9)\), show that \(\cos x=\sqrt{1-\sin ^{2} x}\) for \(0 \leq x \leq\) \(\pi / 2\) and for \(3 \pi / 2 \leq x \leq 2 \pi .\) Show also that \(\cos x=-\sqrt{1-\sin ^{2} x}\) for \(\pi / 2 \leq x \leq 3 \pi / 2\) b. Using \((9)\), show that \(\sin x=\sqrt{1-\cos ^{2} x}\) for \(0 \leq x \leq \pi\) Show also that \(\sin x=-\sqrt{1-\cos ^{2} x}\) for \(\pi \leq x \leq 2 \pi\)

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. $$ x^{2}-4 x+y=5 $$

Solve the inequality. $$ 8 x+\frac{1}{x^{2}}<0 $$

Have a computer or graphics calculator plot the graph of the function on the given \(x\) window. Alter the \(y\) window as needed in order to be able to discern the major features of the graph. By reading off the \(y\) coordinate of the lowest point on the graph, approximate the smallest value assumed by the function on the given interval. It may be helpful to adjust the window. $$ f(x)=\frac{x^{6}-x^{4}+x^{2}+1}{x^{4}+1} ;[-10,10] $$

Find the domain of the function. $$ k(x)=\frac{1}{x+1}-\frac{2}{x-1} $$

Write \(h\) as the composite \(g \circ f\) of two functions \(f\) and \(g\) (neither of which is equal to \(h\) ). $$ h(x)=\frac{1}{\left(x^{3}-2 x^{2}\right)^{5}} $$

Determine at which points the graphs of the given pair of functions intersect. $$ f(x)=\log _{3} x \text { and } g(x)=\log _{2} x $$

Find the tangent of the angle \(\theta\) from the first line to the second line. \(y=4 x-2 ; 3 y=-2 x+7\)

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. \(x=2\) and \(x=-5\)

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. $$ x^{2}+y^{2}+4 x-6 y+13=0 $$

Solve the inequality. $$ \frac{4 x\left(x^{2}-6\right)}{x^{2}-4}<0 $$

Find the domain of the function. $$ f(x)=\left\\{\begin{array}{l} 2 x \text { for }-4 \leq x \leq-1 \\ 3 \text { for } 0

Have a computer or graphics calculator plot the graph of the function on the given \(x\) window. Alter the \(y\) window as needed in order to be able to discern the major features of the graph. By reading off the \(y\) coordinate of the lowest point on the graph, approximate the smallest value assumed by the function on the given interval. It may be helpful to adjust the window. $$ f(x)=210 x^{4}-107 x^{3}+18 x^{2}-x ;[0,0.5] $$

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

Determine at which points the graphs of the given pair of functions intersect. $$ f(x)=3^{x} \text { and } g(x)=2^{\left(x^{3}\right)} \text { (Hint: Take natural logarithms.) } $$

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. \(x=-1\) and \(y=4\)

Find the tangent of the angle \(\theta\) from the first line to the second line. \(y=3 x+9 ; 4 y-11 x=6\)

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. $$ x^{2}-6 x+y^{2}-y=-9 $$