Find a rational function that satisfies the given conditions. Answers may​ vary, but try to give the simplest answer possible.

Vertical asymptotes 

xequals=negative 1−1​, 

xequals=77​; 

​x-intercept 

​(22​,0)

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Part 1

A.

​f(x)equals=StartFraction x plus 2 Over x squared minus 6 x minus 7 EndFractionx+2x2−6x−7

B.

​f(x)equals=StartFraction x minus 2 Over x squared plus 6 x plus 7 EndFractionx−2x2+6x+7

C.

​f(x)equals=StartFraction x minus 2 Over x squared minus 6 x minus 7 EndFractionx−2x2−6x−7

D.

​f(x)equals=StartFraction x plus 2 Over x squared minus 6 x plus 7 EndFraction

We are asked to find a rational function that satisfies the following conditions:

- **Vertical asymptotes**: \( x = -1 \) and \( x = 7 \)
- **x-intercept**: \( (2, 0) \)

### Step 1: General form of a rational function
A rational function with vertical asymptotes at \( x = -1 \) and \( x = 7 \) would have a denominator of the form \( (x + 1)(x - 7) \), because the denominator must be zero at \( x = -1 \) and \( x = 7 \).

Thus, the general form of the function can be written as:
\[
f(x) = \frac{N(x)}{(x + 1)(x - 7)}
\]
where \( N(x) \) is the numerator that we will determine next.

### Step 2: Determining the numerator
We are also given that the function has an x-intercept at \( (2, 0) \), meaning \( f(2) = 0 \). This implies that \( x - 2 \) must be a factor of the numerator \( N(x) \). Therefore, we have:
\[
N(x) = x - 2
\]

### Step 3: Final function
Thus, the simplest rational function that satisfies the given conditions is:
\[
f(x) = \frac{x - 2}{(x + 1)(x - 7)}
\]

### Step 4: Match with options
Now, comparing this with the provided options:

- **Option A**: \( f(x) = \frac{x + 2}{x^2 - 6x - 7} \) — incorrect, wrong numerator.
- **Option B**: \( f(x) = \frac{x - 2}{x^2 + 6x + 7} \) — incorrect, wrong denominator.
- **Option C**: \( f(x) = \frac{x - 2}{x^2 - 6x - 7} \) — correct, because \( x^2 - 6x - 7 \) factors into \( (x + 1)(x - 7) \).
- **Option D**: \( f(x) = \frac{x + 2}{x^2 - 6x + 7} \) — incorrect, wrong numerator and denominator.

The correct answer is **Option C**: \( f(x) = \frac{x - 2}{x^2 - 6x - 7} \).

Would you like more details or have any questions?

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Here are 5 related questions:
1. How do you determine the vertical asymptotes of a rational function?
2. How can you find the x-intercepts of a rational function?
3. What is the significance of factoring the denominator in rational functions?
4. How do horizontal asymptotes differ from vertical asymptotes in rational functions?
5. Can a rational function have more than one x-intercept?

**Tip:** Always check the factors of both the numerator and denominator to determine intercepts and asymptotes of rational functions.

Find a rational function that satisfies the given conditions. Vertical asymptotes: x = -1, x = 7; x-intercept (2,0). Which of the following is the correct function? A) f(x) = (x + 2) / (x^2 - 6x - 7) B) f(x) = (x - 2) / (x^2 + 6x + 7) C) f(x) = (x - 2) / (x^2 - 6x - 7) D) f(x) = (x + 2) / (x^2 - 6x + 7)

Learn how to find a rational function with vertical asymptotes at x = -1, x = 7, and an x-intercept at (2,0). This detailed solution covers core algebraic concepts such as vertical asymptotes, factoring quadratics, and rational functions. Suitable for students in Grades 10-12.

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