Question

let (f(x) = -x^2 + 3x + 1) and (g(x) = 2x).

look at the composition shown. what is the error in the students work?

(f(g(x)) = -(2x)^2 + 3x + 1)

(= -4x^2 + 3x + 1)

Explanation:

⚡ Using what you learned: Evaluating Composite Functions

Step 1: Analyze the functions and the student's work

We are given:
\[ f(x) = -x^2 + 3x + 1 \]
\[ g(x) = 2x \]

The student attempted to find the composite function \( f(g(x)) \):
\[ f(g(x)) = -(2x)^2 + 3x + 1 \]
\[ = -4x^2 + 3x + 1 \]

Step 2: Identify the error

To find \( f(g(x)) \), we must substitute the entire expression for \( g(x) \), which is \( 2x \), into every instance of \( x \) in \( f(x) \).

• The student substituted \( 2x \) into the first term: \( -(2x)^2 \)
• The student failed to substitute \( 2x \) into the second term, leaving it as \( 3x \) instead of \( 3(2x) \).

Step 3: Determine the correct expression

Substitute \( 2x \) for every \( x \) in \( f(x) \):
\[ f(g(x)) = -(2x)^2 + 3(2x) + 1 \]
\[ f(g(x)) = -4x^2 + 6x + 1 \]

Answer:

The student did not substitute \( g(x) \) into the second term of \( f(x) \). They only substituted \( 2x \) for the \( x^2 \) term and left the \( 3x \) term unchanged. The correct expression is \( -4x^2 + 6x + 1 \).