Question

If $f(x) = x^2 + 1$ and $g(x) = 2x - 3$, then $(f \circ g)(x)$ is equal to:

• A. $4x^2 + 9$
• B. $4x^2 - 12x + 10$
• C. $4x^2 - 12x + 13$
• D. $2x^2 - 6x + 10$

Hint:

Always put parentheses around the inner function when substituting! It prevents mistakes with signs and exponents, especially when expanding binomials like $(2x-3)^2$.

Answer 1

The Correct Option is B

Concept:
• Function composition \((fog)(x)\) means evaluating the function \(f\) at the output of function \(g\).
• Mathematically: \((fog)(x) = f(g(x))\).

Step 1: Substitute \(g(x)\) into the expression
We are given \(g(x) = 2x - 3\).
Therefore, \((fog)(x) = f(2x - 3)\).

Step 2: Apply the definition of \(f(x)\)
The function \(f\) is defined as \(f(\text{input}) = (\text{input})^2 + 1\).
Substituting our input \((2x - 3)\):
\((fog)(x) = (2x - 3)^2 + 1\)

Step 3: Expand the algebraic expression
Using the identity \((a - b)^2 = a^2 - 2ab + b^2\):
\((2x - 3)^2 = (2x)^2 - 2(2x)(3) + (3)^2\)
\((2x - 3)^2 = 4x^2 - 12x + 9\)

Step 4: Add the final constant
\((fog)(x) = (4x^2 - 12x + 9) + 1\)
\((fog)(x) = 4x^2 - 12x + 10\)