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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 7: Q. 234 (page 695)

For rational function, $f (x) = \frac{5 - x}{x^{2} + 5 x + 6}$ ,
a) find the domain of the function
b) solve $f (x) = 3$ .
c) find the points on the graph at this function value.

Short Answer

Expert verified

a) The domain is all real numbers except $- 2$ and $- 3$ .

b) The solutions arerole="math" localid="1645288555788" $x = - 1, - \frac{13}{3}$ .

c) The points are $(- 1, 3), (- \frac{13}{3}, 3)$ .

Step by step solution

01

(a)Step 1. Given

The given function is $f (x) = \frac{5 - x}{x^{2} + 5 x + 6}$ .

02

Step 2. To find

Find the domain of the function.

03

Step 3. Calculation

For the domain, the denominator can not be 0.

$x^{2} + 5 x + 6 鈮� 0 (x + 2) (x + 3) 鈮� 0 x 鈮� - 2, - 3$

So the domain is all real numbers except $- 2$ and $- 3$ .

04

(b)Step 1. Given

The given function is $f (x) = \frac{5 - x}{x^{2} + 5 x + 6}$ .

05

Step 2. To find

Solve $f (x) = 3$

06

Step 3. Calculation

Set the given equation equal to 3 and solve for x.

$\frac{5 - x}{x^{2} + 5 x + 6} = 3 5 - x = 3 (x^{2} + 5 x + 6) 5 - x = 3 x^{2} + 15 x + 18 3 x^{2} + 16 x + 13 = 0 3 x^{2} + 3 x + 13 x + 13 = 0 3 x (x + 1) + 13 (x + 1) = 0 (x + 1) (3 x + 13) = 0 x = - 1, - \frac{13}{3}$

07

(c)Step 1. Given

The given function is $f (x) = \frac{5 - x}{x^{2} + 5 x + 6}$ .

08

Step 2. To find

Find the points on the graph at this function value.

09

Step 3. Calculation

The value of $x$ is $x = - 1, - \frac{13}{3}$ .

The value of $y$ is 3.

So, the points are $(- 1, 3), (- \frac{13}{3}, 3)$ .

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Most popular questions from this chapter

In the following exercises, divide the rational expressions.

$\frac{\frac{4 d^{2} + 7 d - 2}{35 d + 10}}{\frac{d^{2} - 4}{7 d^{2} - 12 d - 4}}$

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$\frac{v - 5}{11 - v} 梅 \frac{v^{2} - 25}{v - 11}$

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