Preparing for Calculus

P.1 Assess Your Understanding
Printed Page 12

Concepts and Vocabulary

If $f$ is a function defined by $y=f ( x )$ , then $x$ is called the _____ variable and $y$ is the _____ variable.

Independent, dependent

True or False The independent variable is sometimes referred to as the argument of the function.

True

True or False If no domain is specified for a function $f,$ then the domain of $f$ is taken to be the set of all real numbers.

False

True or False The domain of the function $f ( x ) =\dfrac{3 ( x^{2}-1 ) }{x-1}$ is $\{ x|x\neq \pm 1 \} .$

False

True or False A function can have more than one $y$ -intercept.

False

A set of points in the $xy$ -plane is the graph of a function if and only if every _____ line intersects the graph in at most one point.

Vertical

If the point $( 5,-3 )$ is on the graph of $f$ , then $f($ _____ $)$ $=$ _____.

$5,-3$

Find $a$ so that the point $( -1,2 )$ is on the graph of $f( x ) =ax^{2}+4$ .

$-2$

Multiple Choice A function $f$ is [(a) increasing, (b) decreasing, (c) nonincreasing, (d) nondecreasing, (e) constant] on an interval $I$ if, for any choice of $x_{1}$ and $x_{2}$ in $I,$ with $x_{1}<x_{2}$ , then $f( x_{1}) <f( x_{2})$ .

(a)

Multiple Choice A function $f$ is [(a) even, (b) odd, (c) neither even nor odd] if for every number $x$ in its domain, the number $-x$ is also in the domain and $f( -x) =f( x)$ . A function $f$ is [(a) even, (b) odd, (c) neither even nor odd] if for every number $x$ in its domain, the number $-x$ is also in the domain and $f( -x) =-f( x)$ .

(a), (b)

True or False Even functions have graphs that are symmetric with respect to the origin.

False

The average rate of change of $f( x) =2x^{3}-3$ from $0$ to $2$ is _____.

Practice Problems

In Problems 13–16, for each function find:

$f ( 0)$
$f ( -x)$
$-f ( x)$
$f (x+1)$
$f ( x+h)$

$f( x) =3x^{2}+2x-4$

$-4$
$3x^2-2x-4$
$-3x^2-2x+4$
$3x^2+8x+1$
$3x^2+6xh+3h^2+2x+2h-4$

$f( x) =\dfrac{x}{x^{2}+1}$

$f(x) =\vert x\vert +4$

$4$
$|x|+4$
$-|x|-4$
$|x+1|+4$
$|x+h|+4$

$f( x) = \sqrt{3-x}$

In Problems 17–22, find the domain of each function.

$f( x) =x^{3}-1$

$(-\infty, \infty)$

$f( x) =\dfrac{x}{x^{2}+1}$

$v( t) = \sqrt{t^{2}-9}$

$(-\infty, -3] \cup [3, \infty)$

$g( x) = \sqrt{\dfrac{2}{x-1}}$

$h( x) =\dfrac{x+2}{x^{3}-4x}$

$\{x | x \neq -2, 0, 2\}$

$s( t) =\dfrac{ \sqrt{t+1}}{t-5}$

In Problems 23–28, find the difference quotient of f. That is, find $\dfrac{f ( x+h) -f ( x) }{h}$ , $h\neq 0$ .

$f ( x) =-3x+1$

$-3$

$f ( x) =\dfrac{1}{x+3}$

$f ( x) = \sqrt{x+7}$

$\dfrac{1}{\sqrt{x+h+7}+\sqrt{x+7}}$

$f ( x) =\dfrac{2}{ \sqrt{x+7}}$

$f ( x) =x^{2}+2x$

$2x+h+2$

$f ( x) = ( 2x+3) ^{2}$

In Problems 29–32, determine whether the graph is that of a function by using the Vertical-line Test. If it is, use the graph to find

(a) the domain and range
(b) the intercepts, if any
(c) any symmetry with respect to the x-axis, y-axis, or the origin.

It is not a function.

Domain: $[-\pi, \pi]$ , range: $[-1, 1]$
Intercepts: $\left(-\frac{\pi}{2},0\right), \left(\frac{\pi}{2},0\right), (0,1)$
Symmetric with respect to the $y$ -axis, but not with respect to the $x$ -axis or the origin.

In Problems 33–36, for each piecewise-defined function:

Find $f ( -1 )$ , $f ( 0 )$ , $f ( 1 )$ and $f ( 8 )$ .
Graph $f$ .
Find the domain, range, and intercepts of $f.$

$f( x) =\left\{ \begin{array}{l@{\qquad}l@{\quad}l} x+3 & {\rm if}& -2\leq x<1 \\ 5 & {\rm if}& x=1 \\ -x+2 & {\rm if}& x>1 \end{array} \right.$

$f(-1)=2$ , $f(0) = 3$ , $f(1) = 5$ , $f(8) = -6$
(b)
Domain: $[-2, \infty)$ , range: $(-\infty, 4)\cup \{5\}$ , intercepts: $(0,3)$ , $(2,0)$

$f( x) =\left\{ \begin{array}{l@{\qquad}l@{\quad}l} 2x+5 & {\rm if} & -3\leq x<0 \\ -3 & {\rm if} & x=0 \\ -5x & {\rm if} & x>0 \end{array} \right.$

$f( x) =\left\{ \begin{array}{l@{\qquad}l@{\quad}l} 1+x & {\rm if} & x<0 \\ x^{2} & {\rm if} & x\geq 0 \end{array} \right.$

$f(-1)= 0$ , $f(0) = 0$ , $f(1) = 1$ , $f(8) = 64$
(b)
Domain: $(-\infty, \infty)$ , range: $(-\infty, \infty)$ , intercepts: $(-1,0)$ , $(0,0)$

$f( x) =\left\{ \begin{array}{l@{\qquad}l@{\quad}l} \dfrac{1}{x} & {\rm if} & x<0 \\ \sqrt[3]{x} & {\rm if} & x\geq 0 \end{array} \right.$

In Problems 37–54, use the graph of the function $f$ to answer the following questions.

Find $f ( 0)$ and $f( -6)$ .

$f(0)=3$ , $f(-6)=-3$

Is $f ( 3)$ positive or negative?

Is $f ( -4)$ positive or negative?

Negative

For what values of $x$ is $f ( x) =0$ ?

For what values of $x$ is $f ( x) >0$ ?

$(-3, 6) \cup (10, 11]$

What is the domain of $f$ ?

What is the range of $f$ ?

$[-3, 3]$

What are the $x$ -intercepts?

What is the $y$ -intercept?

How often does the line $y=\dfrac{1}{2}$ intersect the graph?

How often does the line $x=5$ intersect the graph?

Once

For what values of $x$ does $f( x) =3$ ?

For what values of $x$ does $f( x) =-2$ ?

$-5$ , $8$

On what interval(s) is the function $f$ increasing?

On what interval(s) is the function $f$ decreasing?

$(4, 8)$

On what interval(s) is the function $f$ constant?

On what interval(s) is the function $f$ nonincreasing?

$(0, 8)$

On what interval(s) is the function $f$ nondecreasing?

In Problems 55–60, answer the questions about the function $g( x) =\dfrac{x+2}{x-6}.$

What is the domain of $g$ ?

$\{x | x \neq 6\}$

Is the point $( 3,14)$ on the graph of $g$ ?

If $x=4$ , what is $g( x)$ ? What is the corresponding point on the graph of $g$ ?

$-3$ , $(4, -3)$

If $g( x) =2$ , what is $x$ ? What is(are) the corresponding point(s) on the graph of $g$ ?

List the $x$ -intercepts, if any, of the graph of $g$ .

$-2$

What is the $y$ -intercept, if there is one, of the graph of $g$ ?

In Problems 61–64, determine whether the function is even, odd, or neither. Then determine whether its graph is symmetric with respect to the $y$ -axis, the origin, or neither.

$h( x) =\dfrac{x}{x^{2}-1}$

Odd; symmetric with respect to the origin, but not with respect to the $x$ -axis or the $y$ -axis.

$f( x) =\sqrt[3]{3x^{2}+1}$

$G( x) = \sqrt{x}$

Neither; not symmetric to the $x$ -axis, $y$ -axis, or the origin.

$F( x) =\dfrac{2x}{\vert x\vert }$

Find the average rate of change of $f(x) =-2x^{2}+4$ :
1. From $1$ to $2$
2. From $1$ to $3$
3. From $1$ to $4$
4. From $1$ to $x$ , $x\neq 1$

$-4$
$-8$
$-10$
$-2(x+1)$

Find the average rate of change of $s( t) =20-0.8t^{2}$ :
1. From $1$ to $4$
2. From $1$ to $3$
3. From $1$ to $2$
4. From $1$ to $t$ , $t\neq 1$

In Problems 67–72, the graph of a piecewise-defined function is given. Write a definition for each piecewise-defined function. Then state the domain and the range of the function.

$f(x) = \left\{\begin{array}{ll@{\quad}l} -x&\hbox{ if} & -1 \le x < 0\\ \frac{1}{2}x&\hbox{ if} & 0 \le x \le 2 \end{array}\right.\!$ domain: $[-1, 2]$ , range: $[0, 1]$

$f(x) = \left\{\begin{array}{ll@{\quad}l} -1&\hbox{ if} & x < 1\\ 0&\hbox{ if} & x = 1\\ 2-x&\hbox{ if} & 1 < x \le 2 \end{array}\right.$ domain: $(-\infty, 2]$ , range: $\{-1\}\,{\cup}\,[0, 1)$

$\$250.60$
$\$2655.40$
(c) Answers will vary.

The monthly cost $C$ , in dollars, of manufacturing $x$ road bikes is given by the function $C( x) =0.004x^{3}-0.6x^{2}+250x+100{,}500$
1. Find the average rate of change of the cost $C$ of manufacturing from $100$ to $101$ road bikes.
2. Find the average rate of change of the cost $C$ of manufacturing from $500$ to $501$ road bikes.
3. (c) Interpret the results from parts (a) and (b).

The weekly cost in dollars to produce $x$ tons of steel is given by the function $C( x) =\dfrac{1}{10}x^{2}+5x+1500$
1. Find the average rate of change of the cost $C$ of producing from $500$ to $501$ tons of steel.
2. Find the average rate of change of the cost $C$ of producing from $1000$ to $1001$ tons of steel.
3. (c) Interpret the results from parts (a) and (b).

P.1 Assess Your UnderstandingPrinted Page 12

P.1 Assess Your Understanding
Printed Page 12