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 \( 2, 0 \).
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