Question

Which graph represents the function of f(x) = the quantity of 4 x squared minus 4 x minus 8, all over 2 x plus 2?

graph of 2 x minus 4, with discontinuity at negative 1, negative 6
graph of 2 x minus 4, with discontinuity at 1, negative 2
graph of 2 x plus 2, with discontinuity at negative 1, 0
graph of 2 x plus 2, with discontinuity at 1, 4

Answer 1

To analyze the function \( f(x) = \frac{4x^2 - 4x - 8}{2x + 2} \), we will first simplify it.

1. **Factor the numerator:**
\[
4x^2 - 4x - 8 = 4(x^2 - x - 2)
\]
The quadratic \( x^2 - x - 2 \) can be factored further:
\[
x^2 - x - 2 = (x - 2)(x + 1)
\]
Therefore, the numerator becomes:
\[
4(x - 2)(x + 1)
\]

2. **Factor the denominator:**
The denominator can be factored as:
\[
2x + 2 = 2(x + 1)
\]

3. **Rewrite the function:**
Now we can write \( f(x) \) as:
\[
f(x) = \frac{4(x - 2)(x + 1)}{2(x + 1)}
\]
For \( x \neq -1 \), we can simplify this to:
\[
f(x) = \frac{4(x - 2)}{2} = 2(x - 2) = 2x - 4
\]

4. **Identify the discontinuity:**
The function \( f(x) = 2x - 4 \) is defined everywhere except at \( x = -1 \). At \( x = -1 \), the original function has a removable discontinuity since the term \( (x + 1) \) cancels out.

5. **Find the value of \( f(-1) \)**:
To find the value of the function at the discontinuity:
\[
f(-1) = 2(-1) - 4 = -2 - 4 = -6
\]

Therefore, the graph of \( f(x) = 2x - 4 \) has a discontinuity at \( x = -1 \), with the point of discontinuity being \( (-1, -6) \).

Thus, the correct answer is **the graph of \( 2x - 4 \), with discontinuity at \( (-1, -6) \)**.