Questions: Express your answer as a polynomial in standard form. f(x)=x^2+5x-5 g(x)=-3x-7 Find: (f circ g)(x)

Transcript text: Express your answer as a polynomial in standard form. \[ \begin{array}{l} f(x)=x^{2}+5 x-5 \\ g(x)=-3 x-7 \end{array} \] Find: $(f \circ g)(x)$

Solution

Solution Steps

To find $(f \circ g)(x)$, we need to substitute $g(x)$ into $f(x)$. This means wherever there is an $x$ in $f(x)$, we replace it with $-3x - 7$. After substitution, simplify the expression to get the polynomial in standard form.

Step 1: Define the Functions

We are given two functions: \[ f(x) = x^2 + 5x - 5 \] \[ g(x) = -3x - 7 \]

Step 2: Substitute $ g(x) $ into $ f(x) $

To find $(f \circ g)(x)$, substitute $ g(x) $ into $ f(x) $: \[ f(g(x)) = f(-3x - 7) = (-3x - 7)^2 + 5(-3x - 7) - 5 \]

Step 3: Simplify the Expression

Expand and simplify the expression: \[ (-3x - 7)^2 = 9x^2 + 42x + 49 \] \[ 5(-3x - 7) = -15x - 35 \]

Combine all terms: \[ f(g(x)) = 9x^2 + 42x + 49 - 15x - 35 - 5 \]

Step 4: Combine Like Terms

Combine the like terms to simplify the polynomial: \[ f(g(x)) = 9x^2 + (42x - 15x) + (49 - 35 - 5) \] \[ f(g(x)) = 9x^2 + 27x + 9 \]