Analyzing Rational Function f(x) = (x + 11) / (x

Math Problem Statement

Answer the questions about the following function. f left parenthesis x right parenthesis equals StartFraction x plus 11 Over x minus 3 EndFraction (a) Is the point left parenthesis 6 comma nine halves right parenthesis on the graph of f? (b) If x = 1 comma what is f(x)? What point is on the graph of f? (c) If f(x) = 2, what is x? What point(s) is/are on the graph of f? (d) What is the domain of f? (e) List the x-intercepts, if any, of the graph of f. (f) List the y-intercept, if there is one, of the graph of f. . . . Question content area right Part 1 (a) Choose the correct answer. A. No, because f left parenthesis nine halves right parenthesis not equals6. B. No, because f(6)not equalsnine halves . C. Yes, because f(6)equalsnine halves . D. Yes, because f left parenthesis nine halves right parenthesis equals6. Part 2 (b) If xequals1 comma f(x)equals

enter your response here. (Simplify your answer.) Part 3 Using the information in the previous step, list the point(s) on the graph of f where xequals1.

enter your response here (Simplify your answer. Type an ordered pair. Use a comma to separate answers as needed.) Part 4 (c) If f left parenthesis x right parenthesis equals 2 comma xequals

enter your response here. (Simplify your answer. Use a comma to separate answers as needed.) Part 5 Using the information in the previous step, list the point(s) on the graph of f where f(x)equals2.

enter your response here (Simplify your answer. Type an ordered pair. Use a comma to separate answers as needed.) Part 6 (d) What is the domain of f? The domain is

enter your response here. (Type your answer in interval notation.) Part 7 (e) List the x-intercept(s), if any, of the graph of f. Select the correct choice and, if necessary, fill in the answer box to complete your choice. A. The x-intercept(s) is/are

enter your response here. (Use a comma to separate answers as needed.) B. There are no x-intercepts. Part 8 (f) List the y-intercept, if any, of the graph of f. Select the correct choice and, if necessary, fill in the answer box to complete your choice. A. The y-intercept(s) is/are

enter your response here. (Use a comma to separate answers as needed) B. There is no y-intercept. (a) Choose the correct answer.(b) If x1 comma f(x)(Simplify your answer.)Using the information in the previous step, list the point(s) on the graph of f where x1.(Simplify your answer. Type an ordered pair. Use a comma to separate answers as needed.)(c) If f left parenthesis x right parenthesis equals 2 comma x(Simplify your answer. Use a comma to separate answers as needed.)Using the information in the previous step, list the point(s) on the graph of f where f(x)2.(Simplify your answer. Type an ordered pair. Use a comma to separate answers as needed.)The domain is(Type your answer in interval notation.))Select the correct choice and, if necessary, fill in the answer box to complete your choice.input field 7Select the correct choice and, if necessary, fill in the answer box to complete your choice.

Solution

Let's solve each part of the problem step-by-step for the function $f(x) = \frac{x + 11}{x - 3}$ .

(a) Is the point $(6, \frac{9}{2})$ on the graph of $f$ ?

To determine if the point $(6, \frac{9}{2})$ is on the graph of $f(x)$ , we need to check if $f(6) = \frac{9}{2}$ .

First, compute $f(6)$ :

$f(6) = \frac{6 + 11}{6 - 3} = \frac{17}{3}$

Since $\frac{17}{3} \neq \frac{9}{2}$ , the point $(6, \frac{9}{2})$ is not on the graph of $f$ .

Answer (a): B. No, because $f(6) \neq \frac{9}{2}$ .

(b) If $x = 1$ , what is $f(x)$ ? What point is on the graph of $f$ ?

Calculate $f(1)$ :

$f(1) = \frac{1 + 11}{1 - 3} = \frac{12}{-2} = -6$

So, when $x = 1$ , $f(x) = -6$ .

The point $(1, -6)$ is on the graph of $f$ .

Answer (b): $f(1) = -6$ , and the point is $(1, -6)$ .

(c) If $f(x) = 2$ , what is $x$ ? What point(s) is/are on the graph of $f$ ?

Set $f(x) = 2$ and solve for $x$ :

$\frac{x + 11}{x - 3} = 2$

Cross-multiply:

$x + 11 = 2(x - 3)$

Expand and simplify:

$x + 11 = 2x - 6 \implies 11 + 6 = 2x - x \implies 17 = x$

So, $x = 17$ .

The point $(17, 2)$ is on the graph of $f$ .

Answer (c): $x = 17$ , and the point is $(17, 2)$ .

(d) What is the domain of $f$ ?

The function $f(x) = \frac{x + 11}{x - 3}$ is defined for all $x$ except where the denominator is zero.

Set the denominator equal to zero to find the restriction:

$x - 3 = 0 \implies x = 3$

So, the domain of $f$ is all real numbers except $x = 3$ .

Answer (d): The domain is $(-\infty, 3) \cup (3, \infty)$ .

(e) List the $x$ -intercepts, if any, of the graph of $f$ .

The $x$ -intercepts occur where $f(x) = 0$ :

$\frac{x + 11}{x - 3} = 0$

The numerator must be zero:

$x + 11 = 0 \implies x = -11$

So, the $x$ -intercept is $(-11, 0)$ .

Answer (e): A. The $x$ -intercept is $(-11, 0)$ .

(f) List the $y$ -intercept, if there is one, of the graph of $f$ .

The $y$ -intercept occurs where $x = 0$ :

$f(0) = \frac{0 + 11}{0 - 3} = \frac{11}{-3} = -\frac{11}{3}$

So, the $y$ -intercept is $(0, -\frac{11}{3})$ .

Answer (f): A. The $y$ -intercept is $(0, -\frac{11}{3})$ .

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:

How do you determine vertical asymptotes for rational functions?
What is the significance of the numerator and the denominator in a rational function?
How do you find horizontal asymptotes for a rational function?
How can you determine if a function is increasing or decreasing on a given interval?
What are some common applications of rational functions in real-world scenarios?

Tip: When working with rational functions, always check for restrictions in the domain by setting the denominator equal to zero!

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Math Problem Analysis

Mathematical Concepts

Rational functions
Domain and range
Intercepts

Formulas

Rational function formula: f(x) = (x + 11) / (x - 3)

Theorems

Suitable Grade Level

High school

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