To solve this problem, let's go through the composition of functions step-by-step. We're given:

1. $g(x) = x - 1$
2. $(f \circ g)(x) = f(g(x)) = 2x^2 - 4x + 3$

Our goal is to find $f(x)$.

Step 1: Substitute $g(x)$ into $f(g(x))$

Since $g(x) = x - 1$, we replace $g(x)$ in $f(g(x))$: $f(x - 1) = 2x^2 - 4x + 3$

Step 2: Determine the function $f(x)$

To find $f(x)$, let $u = x - 1$. Then, $x = u + 1$. Substitute $x = u + 1$ into the expression $2x^2 - 4x + 3$:

1. Substitute $x = u + 1$ into $x^2$: $x^2 = (u + 1)^2 = u^2 + 2u + 1$

2. Substitute $x = u + 1$ into $-4x$: $-4x = -4(u + 1) = -4u - 4$

3. Combine terms to find $f(u)$: $f(u) = 2(u^2 + 2u + 1) - 4u - 4 + 3$ $= 2u^2 + 4u + 2 - 4u - 4 + 3$ $= 2u^2 + 1$

So, $f(x) = 2x^2 + 1$.

Answer

The correct answer is: E. $f(x) = 2x^2 + 1$

Would you like further details or have any questions?

Related Questions

1. How do we solve for $f(x)$ if $(f \circ g)(x)$ has a different form?
2. What are the steps to verify if two functions are inverses of each other?
3. How would the solution change if $g(x)$ was a more complex function?
4. How do we compute the composition of three functions?
5. What are real-world applications of function composition?

Tip

In function composition, always substitute the inner function result first, then simplify.