Question

Which graph represents the function of f(x) = the quantity of 4 x squared minus 16, all over 2 x minus 4?

graph of 2 x plus 4, with discontinuity at negative 2, 0
graph of 2 x plus 4, with discontinuity at 2, 8
graph of 2 x minus 4, with discontinuity at negative 2, negative 8
graph of 2 x minus 4, with discontinuity at 2, 0

Answer 1

To analyze the function \( f(x) = \frac{4x^2 - 16}{2x - 4} \), we first simplify it.

**Step 1: Factor the numerator.**
The numerator \( 4x^2 - 16 \) can be factored as follows:
\[
4x^2 - 16 = 4(x^2 - 4) = 4(x - 2)(x + 2)
\]

**Step 2: Factor the denominator.**
The denominator \( 2x - 4 \) can be factored out as:
\[
2x - 4 = 2(x - 2)
\]

**Step 3: Rewrite the function.**
Now we can rewrite the function:
\[
f(x) = \frac{4(x - 2)(x + 2)}{2(x - 2)}
\]

**Step 4: Simplify.**
We can cancel out the \( (x - 2) \) term (keeping in mind that this introduces a restriction, as \( x \neq 2 \)):
\[
f(x) = 2(x + 2), \quad x \neq 2
\]

So, the simplified function \( f(x) = 2(x + 2) \) is valid for all \( x \) except for \( x = 2 \), where there is a discontinuity.

**Step 5: Identify characteristics.**
The simplified function is a linear function with a slope of 2 and a y-intercept of 4, except it is undefined at \( x = 2 \) (which is a removable discontinuity).

**Conclusion: Choice based on the options:**
- The line **\( f(x) = 2x + 4 \)** has a slope of 2 and a y-intercept of 4.
- The function is discontinuous at \( x = 2 \).

Given the options, the correct representation of the function is:
- **Graph of \( 2x + 4 \), with a discontinuity at \( 2, 0 \).**