Chapter 1

Fill in the blank. A function \(f\) is __________ on an interval when, for any \(x_{1}\) and \(x_{2}\) in the interval, \(x_{1}f\left(x_{2}\right)\)

Two functions \(f\) and \(g\) can be combined by the arithmetic operations of ______ , ______, _____ , and _______ to create new functions.

If \(f\) and \(g\) are functions such that \(f(g(x))=x\) and \(g(f(x))=x,\) then the function \(g\) is the ________ function of \(f,\) and is denoted by ________.

Name three types of rigid transformations.

A relation that assigns to each element x from a set of inputs, or _______ , exactly one element \(y\) in a set of outputs, or _______ , is called a _______ .

For an equation in \(x\) and \(y,\) if substitution of \(a\) for \(x\) and \(b\) for \(y\) satisfics the equation, then the point \((a, b)\) is a _____.

Match each equation with its form. (a) \(A x+B y+C=0\) (i) vertical line (b) \(x=a\) (ii) slope-intercept form (c) \(y=b\) (iii) general form (d) \(y=m x+b\) (iv) point-slope form (e) \(y-y_{1}=m\left(x-x_{1}\right)\) (v) horizontal line

Fill in the blank. A function \(f\) is __________ when, for each \(x\) in the domain of \(f, f(-x)=f(x)\).

The domain of \(f\) is the _________ of \(f^{-1},\) and the _________ of \(f^{-1}\) is the range of \(f\).

Match the rigid transformation of \(y=f(x)\) with the correct representation, where \(c>0\). (a) \(h(x)=f(x)+c\) (b) \(h(x)=f(x)-c\) (c) \(h(x)=f(x-c)\) (d) \(h(x)=f(x+c)\) (i) horizontal shift \(c\) units to the left. (ii) vertical shift \(c\) units upward. (iii) horizontal shift \(c\) units to the right. (iv) vertical shift \(c\) units downward.

For an equation that represents y as a function of \(x,\) the _______ variable is the set of all \(x\) in the domain, and the _______ variable is the set of all \(y\) in the range.

The set of all solution points of an equation is the _____ of the equation.

For a line, the ratio of the change in y to the change in x is called the _______ of the line.

The graph of a function \(f\) is the segment from (1,2) to \((4,5),\) including the endpoints. What is the domain of \(f ?\)

Fill in the blanks.A reflection in the \(x\) -axis of \(y=f(x)\) is represented by \(h(x)=\) ______,while a reflection in the \(y\) -axis of \(y=f(x)\) is represented by \(h(x)=\) ________ .

Can the ordered pairs \((3, 0)\) and \((3, 5)\) represent a function?

Name three common approaches you can use to solve problems mathematically.

Two lines are _______ if and only if their slopes are equal.

A vertical line intersects a graph twice. Does the graph represent a function?

To have an inverse function, a function \(f\) must be __________ \(;\) that is, \(f(a)=f(b)\) implies \(a=b.\)

To find \(g(x+1),\) what do you substitute for \(x\) in the function \(g(x)=3 x-2 ?\)

List the steps for sketching the graph of an equation by point plotting.

What is the relationship between two lines whose slopes are -3 and \(\frac{1}{3} ?\)

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.$$\begin{aligned}&f(x)=x\\\&g(x)=x-4\\\&h(x)=3 x\end{aligned}$$.

Let \(f\) be a function such that \(f(2) \geq f(x)\) for all values of \(x\) in the interval (0,3) Does \(f(2)\) represent a relative minimum or a relative maximum?

Given \(f(x)=x^{2}+1\) and \((f g)(x)=2 x\left(x^{2}+1\right),\) what is \(g(x) ?\)

How many times can a horizontal line intersect the graph of a function that is one-to-one?

Determine whether each point lies on the graph of the equation. \(y=\sqrt{x+4}\) (a) (0,2) (b) (12,4)

Does the domain of the function \(f(x)=\sqrt{1+x}\) include \(x=-2 ?\)

What is the slope of a line that is perpendicular to the line represented by \(x=3 ?\)

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.$$\begin{aligned}&f(x)=\frac{1}{2} x\\\&g(x)=\frac{1}{2} x+2\\\&h(x)=4(x-2)\end{aligned}$$.

Can (1,4) and (2,4) be two ordered pairs of a one-to-one function?

Given \(f(x)=[x],\) in what interval does \(f(x)=5 ?\)

Determine whether each point lies on the graph of the equation. \(y=\sqrt{5-x}\) (a) (1,2) (b) (5,0)

Is the domain of a piecewise-defined function implied or explicitly described?

Give the coordinates of a point on the line whose equation in point-slope form is \(y-(-1)=\frac{1}{4}(x-8)\)

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.$$\begin{aligned}&f(x)=x^{2}\\\&g(x)=x^{2}+2\\\&h(x)=(x-2)^{2}\end{aligned}$$.

Find the inverse function of \(f\) informally. Verify that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x.\) $$f(x)=6 x$$

Determine whether each point lies on the graph of the equation. \(y=4-|x-2|\) (a) (1,5) (b) (1.2,3.2)

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.$$\begin{aligned}&f(x)=x^{2}\\\&g(x)=3 x^{2}\\\&h(x)=(x+2)^{2}+1\end{aligned}$$.

Find the inverse function of \(f\) informally. Verify that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x.\) $$f(x)=\frac{1}{3} x$$

Determine whether each point lies on the graph of the equation. \(y=|x-1|+2\) (a) (2,1) (b) (3.2,4.2)

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.$$\begin{aligned}&f(x)=-x^{2}\\\&g(x)=-x^{2}+1\\\&h(x)=-(x+3)^{2}\end{aligned}$$.

Find the inverse function of \(f\) informally. Verify that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x.\) $$f(x)=x+11$$

Determine whether each point lies on the graph of the equation. \(2 x-y-3=0\) (a) (1,2) (b) (1,-1)

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.$$\begin{aligned}&f(x)=(x-2)^{2}\\\&g(x)=(x+2)^{2}+2\\\&h(x)=-(x-2)^{2}-1\end{aligned}$$.

Find the inverse function of \(f\) informally. Verify that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x.\) $$f(x)=x+3$$

Determine whether each point lies on the graph of the equation. \(x^{2}+y^{2}=20\) (a) (3,-2) (b) (-4,2)

Use a graphing utility to graph the function and estimate its domain and range. Then find the domain and range algebraically. $$f(x)=-2 x^{2}+3$$

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.$$\begin{aligned}&f(x)=x^{2}\\\&g(x)=\frac{1}{2} x^{2}\\\&h(x)=(2 x)^{2}\end{aligned}$$.