Question

If \( f(x) = ax + bx^2 \), then the coefficient of \( x^3 \) in \( f(f(x)) \) is

• A. \( 0 \)
• B. \( ab \)
• C. \( a^3 \)
• D. \( ab^2 \)
• E. \( 2ab^2 \)

Hint:

Whenever you need the coefficient of a particular power in a composite function, first write the composition carefully, then expand only up to the required power. This saves time and reduces mistakes.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This problem involves the composition of a function with itself, denoted as \(f(f(x))\) or \((f \circ f)(x)\). This means we substitute the expression for \(f(x)\) into the variable \(x\) within the function's definition. After finding the composite function, we need to identify the coefficient of the \(x^3\) term.
Step 2: Key Formula or Approach:
Given \(f(x) = ax + bx^2\).
The composite function is found by replacing \(x\) with \(f(x)\):
\[ f(f(x)) = a(f(x)) + b(f(x))^2 \] Step 3: Detailed Explanation:
Substitute the expression \(ax + bx^2\) for \(f(x)\) in the equation above:
\[ f(f(x)) = a(ax + bx^2) + b(ax + bx^2)^2 \] Now, we need to expand this expression and find the term containing \(x^3\).
Expand the first part:
\[ a(ax + bx^2) = a^2x + abx^2 \] This part does not produce an \(x^3\) term.
Expand the second part. We use the identity \((p+q)^2 = p^2 + 2pq + q^2\).
\[ b(ax + bx^2)^2 = b((ax)^2 + 2(ax)(bx^2) + (bx^2)^2) \] \[ = b(a^2x^2 + 2abx^3 + b^2x^4) \] Now, distribute the \(b\) outside the parenthesis:
\[ = a^2bx^2 + 2ab^2x^3 + b^3x^4 \] The term containing \(x^3\) from this part is \(2ab^2x^3\).
Combine both expanded parts:
\[ f(f(x)) = (a^2x + abx^2) + (a^2bx^2 + 2ab^2x^3 + b^3x^4) \] The only term with \(x^3\) in the entire expression is \(2ab^2x^3\).
Step 4: Final Answer:
The coefficient of the \(x^3\) term in the expansion of \(f(f(x))\) is \(2ab^2\). Therefore, option (E) is the correct answer.