Question

If \[ f(x) = \frac{1}{x^2 + 4x + 4} - \frac{4}{x^4 + 4x^3 + 4x^2} + \frac{4}{x^3 + 2x^2}, \] then the value of \[ f\left(\frac{1}{2}\right) \] is equal to:

• A. \( 1 \)
• B. \( 2 \)
• C. \( -1 \)
• D. \( 3 \)
• E. \( 4 \)

Hint:

Directly substituting \( 1/2 \) into the original expression would lead to messy fractions. Algebraic simplification is almost always the intended first step in such function problems.

Answer 1

Answer: (E)

Concept: To find the value of a complex rational function, it is often easier to simplify the algebraic expression using factoring and a common denominator before substituting the numerical value.

Step 1: Factoring the denominators.
1) \( x^2 + 4x + 4 = (x+2)^2 \)
2) \( x^4 + 4x^3 + 4x^2 = x^2(x^2 + 4x + 4) = x^2(x+2)^2 \)
3) \( x^3 + 2x^2 = x^2(x+2) \)
Now substitute these back: \[ f(x) = \frac{1}{(x+2)^2} - \frac{4}{x^2(x+2)^2} + \frac{4}{x^2(x+2)} \]

Step 2: Simplifying the expression.
Using the common denominator \( x^2(x+2)^2 \): \[ f(x) = \frac{x^2 - 4 + 4(x+2)}{x^2(x+2)^2} = \frac{x^2 - 4 + 4x + 8}{x^2(x+2)^2} \] \[ f(x) = \frac{x^2 + 4x + 4}{x^2(x+2)^2} = \frac{(x+2)^2}{x^2(x+2)^2} = \frac{1}{x^2} \]

Step 3: Evaluating for \( x = 1/2 \).
\[ f\left(\frac{1}{2}\right) = \frac{1}{(1/2)^2} = \frac{1}{1/4} = 4 \]