Question

Express your answer as a polynomial in standard form.
f, of, x, equals, 2, x, minus, 2
f(x)=
2x−2
g, of, x, equals, x, squared, minus, 3, x, minus, 15
g(x)=
x
2
−3x−15
Find: , left parenthesis, g, circle, f, right parenthesis, left parenthesis, x, right parenthesis
Find: (g∘f)(x)

Answer 1

To find \((g \circ f)(x)\), we need to evaluate \(g(f(x))\).

Given:
- \(f(x) = 2x - 2\)
- \(g(x) = x^2 - 3x - 15\)

Now, we substitute \(f(x)\) into \(g(x)\):

\[
g(f(x)) = g(2x - 2)
\]

Next, we replace \(x\) in \(g(x)\) with \(2x - 2\):

\[
g(2x - 2) = (2x - 2)^2 - 3(2x - 2) - 15
\]

Now, let's simplify this step by step.

1. Calculate \((2x - 2)^2\):

\[
(2x - 2)^2 = 4x^2 - 8x + 4
\]

2. Calculate \(-3(2x - 2)\):

\[
-3(2x - 2) = -6x + 6
\]

Now, substitute these results into the expression for \(g(2x - 2)\):

\[
g(2x - 2) = (4x^2 - 8x + 4) + (-6x + 6) - 15
\]

Combine like terms:

\[
g(2x - 2) = 4x^2 - 8x - 6x + 4 + 6 - 15
\]

Combine the \(x\) terms and constants:

\[
= 4x^2 - 14x + (4 + 6 - 15)
\]
\[
= 4x^2 - 14x - 5
\]

Thus, the polynomial in standard form for \((g \circ f)(x)\) is:

\[
\boxed{4x^2 - 14x - 5}
\]