Question

find the indicated intercept(s) of the graph of the function.

x-intercepts of \\(f(x) = \frac{x^2 + 7x}{x^2 + 7x - 2}\\)

a. \\((0,0)\\) and \\((7,0)\\)
b. \\((0,0)\\) and \\((-7,0)\\)
c. \\((-7,0)\\)
d. \\((7,0)\\)

Explanation:

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<reasoning>

Set the function to zero

Using the X-Intercept of Rational Function knowledge point
\[
f(x) = \frac{x^2 + 7x}{x^2 + 7x - 2} = 0
\]

Solve for the numerator zeros

Using the Finding Zeros of Factored Polynomials knowledge point
\[

$$\begin{aligned} &x^2 + 7x = 0 \\ &x(x + 7) = 0 \\ &x = 0 \quad \text{or} \quad x = -7 \end{aligned}$$

\]

Verify domain restrictions

We check if the denominator is non-zero at these values.
For \(x = 0\):
\[
0^2 + 7(0) - 2 = -2
e 0
\]
For \(x = -7\):
\[
(-7)^2 + 7(-7) - 2 = 49 - 49 - 2 = -2
e 0
\]
Both values are valid.

Write as coordinate points

The x-intercepts are:
\[
(0, 0) \quad \text{and} \quad (-7, 0)
\]
</reasoning>

<answer>
<mcq-option>A. (0,0) and (7,0)</mcq-option>
<mcq-correct>B. (0,0) and (-7,0)</mcq-correct>
<mcq-option>C. (-7,0)</mcq-option>
<mcq-option>D. (7,0)</mcq-option>
</answer>

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Answer:

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<reasoning>

Set the function to zero

Using the X-Intercept of Rational Function knowledge point
\[
f(x) = \frac{x^2 + 7x}{x^2 + 7x - 2} = 0
\]

Solve for the numerator zeros

Using the Finding Zeros of Factored Polynomials knowledge point
\[

$$\begin{aligned} &x^2 + 7x = 0 \\ &x(x + 7) = 0 \\ &x = 0 \quad \text{or} \quad x = -7 \end{aligned}$$

\]

Verify domain restrictions

We check if the denominator is non-zero at these values.
For \(x = 0\):
\[
0^2 + 7(0) - 2 = -2
e 0
\]
For \(x = -7\):
\[
(-7)^2 + 7(-7) - 2 = 49 - 49 - 2 = -2
e 0
\]
Both values are valid.

Write as coordinate points

The x-intercepts are:
\[
(0, 0) \quad \text{and} \quad (-7, 0)
\]
</reasoning>

<answer>
<mcq-option>A. (0,0) and (7,0)</mcq-option>
<mcq-correct>B. (0,0) and (-7,0)</mcq-correct>
<mcq-option>C. (-7,0)</mcq-option>
<mcq-option>D. (7,0)</mcq-option>
</answer>

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