1.2.3What happens in the middle?

Holes, Vertical Asymptotes, and Approach Statements

1-44.

For $f(x)=\frac{x^2+5x+3}{x+4}$ and $g(x)=\frac{x^2+5x+4}{x+4}$ :

Draw a careful sketch of each function. Use a dashed line for an asymptote and an open circle for a “hole” (a single point which the graph appears to go through, but where it is actually undefined).
For both $f$ and $g$ , write the equations of all asymptotes and the coordinates of any holes.
State the domains and ranges of $f$ and $g$ .

1-45.

HOLES AND ASYMPTOTES

With your team, write a conjecture that states which rational functions of the form $f(x) =\frac { p ( x ) } { q ( x ) }$ have vertical asymptotes and which have holes. When using your graphing calculator, be sure to use a “friendly window”. To help you get started, several rational functions are given below. Be sure to generate your own rational functions to confirm your conjecture.

Possible rational expressions:

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

$g(x) =\frac { x ^ { 2 } + 2 x + 2 } { x + 1 }$

$h(x) =\frac { x ^ { 2 } - 5 x + 6 } { x - 2 }$

$j(x) =\frac { x ^ { 2 } - 5 x + 6 } { x - 1 }$

1-46.

Approach Statements

A statement that describes the behavior of a function at various locations is called an approach statement. Although you can use shorthand symbols, such as “ $→$ ,” you need to write a complete set of statements involving all of the important “approaches” of the function.

A complete set of approach statements includes the extremes of the domain, as well as any holes or asymptotes. Below is a complete set of approach statements for $y =\frac { 1 } { x - 2 }+3$ .

As $x$ approaches positive infinity, $y$ approaches $3$ .

As $x → ∞, y → 3$ .

Decreasing rational function, asymptotes: y = 3, &, x = 2, left section, opens down, is below & left of asymptote intersection, Right section, opens up, is above & right of asymptote intersection, & labeled with ray pointing right, & x, right arrow, infinity.

As $x$ approaches negative infinity, $y$ approaches $3$ .

As $x → −∞, y → 3$ .

Decreasing rational function, asymptotes: y = 3, &, x = 2, left section, opens down, is below & left of asymptote intersection, & labeled with ray pointing left, & x, right arrow, negative infinity, Right section, opens up, is above & right of asymptote intersection.

As $x$ approaches $2$ from the left, $y$ approaches negative infinity.

As $x → 2^{−}, y → −∞$ .

Decreasing rational function, asymptotes: y = 3, &, x = 2, left section, opens down, is below & left of asymptote intersection, & labeled with ray pointing right, & x, right arrow, 2 superscript, negative, Right section, opens up, is above & right of asymptote intersection.

As $x$ approaches $2$ from the right, $y$ approaches positive infinity.

As $x → 2^{+}, y → ∞$ .

1-47.

Examine the graph of $y =\frac { 1 } { x + 1 }-2$ at right. Use the graph to answer the questions below.

What does $y$ approach as $x → ∞$ ?
What does $y$ approach as $x → −∞$ ?
What does $y$ approach as $x → − 1^−$ (the symbol “ $\text{#}^-$ ” means approaching from the negative direction, or from the left)?
What does $y$ approach as $x → −1^+$ (from the positive direction, or from the right)?
Name all horizontal and vertical asymptotes.

Decreasing rational function, asymptotes: y = negative 2, &, x = negative 1, left section, opens down, is below & left of asymptote intersection, Right section, opens up, is above & right of asymptote intersection.

1-48.

Sketch a graph of an even function that has a vertical asymptote at $x = 2$ , a hole at $x = -4$ , and as $x → ∞$ , $y → 3$ .

1-49.

In problem 1-46, the numerator and denominator were both polynomials. When this is not the case, factoring is no longer useful. For each function in parts (a) through (f):

If the function is defined at $x = 0$ , state the value at $x = 0$ .
If the function is not defined at $x = 0$ , use your calculator to sketch a graph. Clearly indicate whether the function has a hole or an asymptote at $x = 0$ .

$f(x) =\frac { \operatorname { sin } ( x ) } { x }$

$f(x) =\frac { \operatorname { sin } ^ { 2 } ( x ) } { x }$

$f(x) =\frac { \operatorname { sin } ( x ) } { x ^ { 2 } }$

$f(x) =\frac { \operatorname { cos } ( x ) } { x }$

$f(x) =\frac { 1 - \operatorname { cos } ( x ) } { x - 1 }$

$f(x) =\frac { 1 - \operatorname { cos } ( x ) } { x }$

1-50.

For the following functions, when $2$ is substituted for $x$ , the fraction has the indeterminate form (or undefined form) $\frac { 0 } { 0 }$ . State whether the graphs of the following functions have holes, asymptotes, or neither at $x = 2$ . Explain your answer.

$f(x) =\frac { x - 2 } { x - 2 }$

$g(x) = \frac { ( x - 2 ) ^ { 2 } } { x - 2 }$

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

Sketch a graph and write the equation of a function that looks like $y = x - 2$ with a hole at $x = 4$ .

Review and Preview problems below

1-51.

Analyze the graph of $y=\frac{(x+2)(x-1)}{x-1}$ at right. Homework Help ✎

What does $y$ approach as $x → ∞$ ? What does $y$ approach as $x→ −∞$ ? Describe how your answer can be predicted from the given equation.
What does $y$ approach as $x → 1^−$ ( $1$ from the left)? What does $y$ approach as $x → 1^+$ ( $1$ from the right)? Describe how your answer can be predicted from the given equation.

Graph of a line, passing through the points (negative 2, comma 0), & (0, comma 2), with highlighted point at (1, comma 3).

1-52.

Write the equation of a function that has the following complete set of approach statements. Hint: Start by sketching the graph. 1-52 HW eTool. Homework Help ✎

As $x → 3^+ , y → ∞$ .

As $x → 3^–, y → −∞$ .

As $x → −∞, y → 1$ .

As $x → ∞, y → 1$ .

1-53.

Convert the following domain and range from interval notation to set notation. Then sketch a possible function with the given domain and range. Homework Help ✎

$D=\left(−∞,2\right)\cup\left(2,∞\right)$ $R=\left(−∞,-1\right)\cup\left(-1,∞\right)$

1-54.

On graph paper, sketch the function $g ( x ) = \sqrt { 36 - x ^ { 2 } }$ . Shade the area under the curve for $3 ≤ x ≤ 6$ . 1-54 HW eTool. Homework Help ✎

Use geometry to calculate this area. Hint: Draw in a radius to create two easier regions whose difference is the shaded region.
Calculate the area under the curve for $0 ≤ x ≤ 3$ .
Calculate the area under the curve for $–3 ≤ x ≤ 6$ .

1-55.

A marathon runner runs a $26.2$ -mile race. Her distance traveled in miles after $t$ hours is $p(t) = 7t$ . Homework Help ✎

How long does it take her to finish the race?
What is her average velocity? Explain your reasoning.
Suppose she runs at a constant pace of $7$ miles/hour. How far will she have gone in $2$ hours?
Show how the units in your answer to part (c) reduce using $\text{(rate)(time)} = \text{distance}$ .

1-56.

Wei Kit loves patterns! When making calculations with rational exponents, he looks for a way to avoid using his calculator. For example, he knows that $8^{2/3} = 4$ by using the method below:

$8^{2/3} =( \sqrt [ 3 ] { 8 } ) ^ { 2 }= (2)^2 = 4$

Use Wei Kit’s method to evaluate the following expressions: Homework Help ✎

$100^{3/2}$

$27^{4/3}$

$16^{3/4}$

$9^{4/2}$

1-57.

Sketch a graph of $y = 1 − x^3$ . Then complete the following approach statements. 1-57 HW eTool. Homework Help ✎

As $x → ∞, y$ approaches?
As $x → −∞$ , $y$ approaches?
As $x → 0^−$ ( $0$ from the left), $y$ approaches?

1-58.

If for $f (x) = 2x^2 + 1$ , estimate the area under the curve for $–3 ≤ x ≤ 3$ as follows. Homework Help ✎

Using six left endpoint rectangles. The first two rectangles are drawn for you.
Using six right endpoint rectangles.
Using six trapezoids. What do you notice? Does this always happen?

Upward parabola, vertex at (0, comma 1), & 2 shaded vertical bars, bottom edges on x axis, each with width of 1, starting at x = negative 3, with the top left vertex of each bar, on the parabola.

1-59.

Each of the continuous functions in the table below is increasing, but each increases differently. Match each graph below with the function that grows in a similar fashion in the table. 1-59 HW eTool Homework Help ✎

$x$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$
$f(x)$	$64$	$68.8$	$74.6$	$81.5$	$89.8$	$99.7$	$111.7$	$126$	$143.2$
$g(x)$	$38$	$52$	$66$	$80$	$94$	$108$	$122$	$136$	$150$
$h(x)$	$22$	$42.9$	$57.3$	$68.5$	$77.6$	$85.3$	$92$	$97.9$	$103.1$

1-60.

When the semi-circular flag below is rotated, it has a volume of $\frac{243}{2}\pi$ un³. Homework Help ✎

Describe the resulting three-dimensional figure.
What is the value of $d$ ?
If the diagram is rotated $90^\circ$ and the flag is then rotated about a horizontal pole, will the volume change?

Vertical segment, with shaded semicircle right of segment, with it's diameter on the top 3 fourths of the segment, distance across semi circle labeled, d.

Calculus Third Edition

Approach Statements