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:

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

$$
LATEXBLOCK0
$$

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:

$$ (0, 0) \quad \text{and} \quad (-7, 0) $$

Answer:

• A. (0,0) and (7,0)
• B. (0,0) and (-7,0) (Correct answer)
• C. (-7,0)
• D. (7,0)