Function f(x) = `(x-2)/(x^(2)-5x+6)` is discontinuous at x = …..

Options

A. `-2,-3`
B. `2,3`
C. `-2,3`
D. `2,-3`

Answer

To determine where the function \( f(x) = \frac{x - 2}{x^2 - 5x + 6} \) is discontinuous, we need to find the points where the denominator is equal to zero, as the function will be undefined at those points.

### Step-by-Step Solution:

1. **Identify the Denominator**:
The denominator of the function is \( x^2 - 5x + 6 \).

2. **Set the Denominator to Zero**:
To find the points of discontinuity, we set the denominator equal to zero:
\[ x^2 - 5x + 6 = 0 \]

3. **Factor the Quadratic Equation**:
We need to factor the quadratic equation. We look for two numbers that multiply to \( 6 \) (the constant term) and add up to \( -5 \) (the coefficient of \( x \)):
\[ (x - 2)(x - 3) = 0 \]

4. **Solve for x**:
Now, we set each factor equal to zero:
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]

5. **Identify Points of Discontinuity**:
The function \( f(x) \) is discontinuous at \( x = 2 \) and \( x = 3 \).

### Final Answer:
The function \( f(x) \) is discontinuous at \( x = 2 \) and \( x = 3 \).

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