https://file.aiquickdraw.com/mathbot/file/17266932264184vw2lqpm.jpg

To solve this problem, let's break down the given function and find its discontinuities.

Given:
\[
f(x) = \frac{3x + 4}{x^2 + 4x - 21}
\]

### Step 1: Identify Discontinuities
Discontinuities for a rational function occur where the denominator is zero. So, we need to find the values of \(x\) that make the denominator zero.

The denominator is:
\[
x^2 + 4x - 21 = 0
\]

### Step 2: Solve the Quadratic Equation
To find the roots of the quadratic equation \(x^2 + 4x - 21 = 0\), we use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \(a = 1\), \(b = 4\), and \(c = -21\).

\[
x = \frac{-4 \pm \sqrt{16 + 84}}{2 \times 1} = \frac{-4 \pm \sqrt{100}}{2} = \frac{-4 \pm 10}{2}
\]

This gives us two solutions:
1. \(x = \frac{-4 + 10}{2} = 3\)
2. \(x = \frac{-4 - 10}{2} = -7\)

### Step 3: Determine the Values of Discontinuities
The function \(f(x)\) is discontinuous at \(x = 3\) and \(x = -7\) because these values make the denominator zero.

### Answer
The values of \(x\) where the function has discontinuities are:
\[
x = 3, -7
\]

So, you should enter:
```
3, -7
```

Would you like a more detailed explanation or have any questions?

### Relative Questions
1. What types of discontinuities exist in rational functions?
2. How can you simplify a rational function to identify removable discontinuities?
3. What is the significance of the numerator in determining discontinuities?
4. How do you check the limits around a discontinuity to classify it?
5. Can a rational function have discontinuities where the numerator is zero?

### Tip
Always factor the denominator completely to identify all possible points of discontinuity in a rational function.

Consider the function f(x) = (3x + 4) / (x^2 + 4x - 21). Determine the number of discontinuities and the values of x at which f(x) has discontinuities.

Solve the rational function f(x) = (3x + 4) / (x^2 + 4x - 21) to find its discontinuities. This problem involves solving a quadratic equation to determine the values of x that make the denominator zero. Detailed step-by-step solution included. Suitable for Grades 11-12 or introductory calculus students.

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