The point at which the function `f(x) = (x+1)/(x^(2)+x-12)` is discontinuous are

Options

A. `-3,4`
B. `3,-4`
C. `3,4`
D. `-3,-4`

Answer

To determine the points at which the function \( f(x) = \frac{x+1}{x^2+x-12} \) is discontinuous, we need to find where the denominator is equal to zero, as this will make the function undefined.

### Step-by-Step Solution:

1. **Identify the Denominator**:
The denominator of the function is \( x^2 + x - 12 \).

2. **Set the Denominator to Zero**:
To find the points of discontinuity, we set the denominator equal to zero:
\[ x^2 + x - 12 = 0 \]

3. **Factor the Quadratic Equation**:
We need to factor the quadratic expression. We look for two numbers that multiply to \(-12\) (the constant term) and add up to \(1\) (the coefficient of \(x\)). The numbers \(4\) and \(-3\) satisfy this:
\[ x^2 + 4x - 3x - 12 = 0 \]
This can be grouped as:
\[ (x^2 + 4x) + (-3x - 12) = 0 \]
Factoring by grouping gives:
\[ x(x + 4) - 3(x + 4) = 0 \]
Therefore, we can factor it as:
\[ (x + 4)(x - 3) = 0 \]

4. **Solve for x**:
Set each factor equal to zero:
\[ x + 4 = 0 \quad \text{or} \quad x - 3 = 0 \]
This gives us:
\[ x = -4 \quad \text{and} \quad x = 3 \]

5. **Conclusion**:
The function \( f(x) \) is discontinuous at the points \( x = -4 \) and \( x = 3 \).

### Final Answer:
The points at which the function \( f(x) = \frac{x+1}{x^2+x-12} \) is discontinuous are \( x = -4 \) and \( x = 3 \).

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