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

The notation $(f \circ g)(x)$ represents the composition of functions, which means we must evaluate $f$ at the value of $g(x)$. Essentially, we substitute the entire expression of $g(x)$ into every instance of $x$ in $f(x)$. 1. Setting up the Substitution: Given: $f(x) = x^2 + 1$ $g(x) = 2x - 3$ We need to find $f(g(x))$: $f(g(x)) = (g(x))^2 + 1$ 2. Expanding the Expression: Substitute the expression for $g(x)$: $f(g(x)) = (2x - 3)^2 + 1$ Now, expand the binomial $(a - b)^2 = a^2 - 2ab + b^2$: $(2x - 3)^2 = (2x)^2 - 2(2x)(3) + (3)^2$ $(2x - 3)^2 = 4x^2 - 12x + 9$ 3. Final Simplification: Add the constant $+1$ from the original $f(x)$ function: $(f \circ g)(x) = (4x^2 - 12x + 9) + 1$ $(f \circ g)(x) = 4x^2 - 12x + 10$ Comparing this result to the options, we find it matches Option (2).