Express your answer as a polynomial in standard form.

Question

f, of, x, equals, x, squared, plus, 4, x, minus, 7
f(x)=
x
2
+4x−7
g, of, x, equals, minus, x, minus, 8
g(x)=
−x−8
Find: , left parenthesis, f, circle, g, right parenthesis, left parenthesis, x, right parenthesis
Find: (f∘g)(x)

Answer 1

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

We have:

• \(f(x) = x^2 + 4x - 7\)
• \(g(x) = -x - 8\)

Now, we will compute \(f(g(x))\):

\[ f(g(x)) = f(-x - 8) \]

Substituting \(-x - 8\) into \(f(x)\):

\[ f(-x - 8) = (-x - 8)^2 + 4(-x - 8) - 7 \]

Now, we calculate each term:

1. Calculate \((-x - 8)^2\):

\[ (-x - 8)^2 = (-x)^2 + 2(-x)(-8) + (-8)^2 = x^2 + 16x + 64 \]

2. Calculate \(4(-x - 8)\):

\[ 4(-x - 8) = -4x - 32 \]

Putting these two parts into our expression for \(f(-x - 8)\):

\[ f(-x - 8) = (x^2 + 16x + 64) + (-4x - 32) - 7 \]

Now, combine like terms:

\[ f(-x - 8) = x^2 + 16x - 4x + 64 - 32 - 7 \]

This simplifies to:

\[ f(-x - 8) = x^2 + 12x + 25 \]

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

\[ \boxed{x^2 + 12x + 25} \]