https://file.aiquickdraw.com/mathbot/file/1727065334555o93ok0zx.jpg

Let's work through the problem step by step to identify the issue.

### The Given Functions:
1. \( s(x) = \frac{x - 5}{x^2 - 49} \)
2. \( v(x) = \sqrt{x + 7} \)

### The Requested Function:
We are asked to find the domain of \( \frac{v(x)}{s(x)} \), which is:

\[
\frac{v(x)}{s(x)} = \frac{\sqrt{x + 7} (x - 7)(x + 7)}{x - 5}
\]

### Step 1: Domain of \( v(x) = \sqrt{x + 7} \)
The square root function is defined for values where the expression under the square root is non-negative:

\[
x + 7 \geq 0 \implies x \geq -7
\]

Thus, the domain of \( v(x) \) is \( x \geq -7 \), or in interval notation: \( [-7, \infty) \).

### Step 2: Domain of \( s(x) = \frac{x - 5}{x^2 - 49} \)
The denominator \( x^2 - 49 \) can be factored:

\[
x^2 - 49 = (x - 7)(x + 7)
\]

To avoid division by zero, we must exclude values where the denominator equals zero:

\[
x^2 - 49 = 0 \implies (x - 7)(x + 7) = 0 \implies x = 7 \text{ or } x = -7
\]

Thus, the domain of \( s(x) \) excludes \( x = 7 \) and \( x = -7 \). The domain of \( s(x) \) is:

\[
(-\infty, -7) \cup (-7, 7) \cup (7, \infty)
\]

### Step 3: Combined Domain of \( \frac{v(x)}{s(x)} \)
To find the domain of \( \frac{v(x)}{s(x)} \), we must:
1. Take the domain where \( v(x) \) is defined: \( [-7, \infty) \).
2. Exclude the points where \( s(x) \) causes division by zero, i.e., \( x = -7 \) and \( x = 7 \).

Thus, the domain is:

\[
(-7, 7) \cup (7, \infty)
\]

### The Issue:
In the screenshot, the domain is given as \( [-7, 7) \cup (7, \infty) \), but this is incorrect because \( x = -7 \) should be excluded (since \( s(x) \) is undefined at \( x = -7 \)).

### Correct Domain:
The correct domain should be:

\[
(-7, 7) \cup (7, \infty)
\]

### Summary:
The mistake was including \( -7 \) in the domain. It should be excluded because it causes the denominator of \( s(x) \) to be zero.

What is wrong with the given domain for the function v(x)/s(x)?

This problem demonstrates how to find the domain of the function v(x)/s(x), involving rational expressions and square roots. The key concepts include excluding points where the denominator is zero and where square roots are undefined. Suitable for students in Grades 9-12.

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