Express your answer as a polynomial in standard form.

Question

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:

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