Questions: What is the value of g(f(x)) when x=-2 if f(x)=x^3 and g(x)= 2x+1 when x <= 0 x^2-1 when x>0 -15 -8 0 9 17

Transcript text: What is the value of $g(f(x))$ when $x=-2$ if $f(x)=x^{3}$ and $g(x)=\{$ $2 x+1 \quad$ when $\mathrm{x} \leq 0$ $x^{2}-1 \quad$ when $\mathrm{x}>0$ $-15$ $-8$ 0 9 17

Solution

Solution Steps

Step 1: Evaluate $ f(x) $ at $ x = -2 $

First, we need to find the value of $ f(x) $ when $ x = -2 $. The function $ f(x) = x^3 $.

\[ f(-2) = (-2)^3 = -8 \]

Step 2: Determine which case of $ g(x) $ to use

Next, we need to evaluate $ g(f(x)) $, which means we need to find $ g(-8) $. Since $-8 \leq 0$, we use the first case of the piecewise function for $ g(x) $:

\[ g(x) = 2x + 1 \quad \text{when } x \leq 0 \]

Step 3: Evaluate $ g(f(x)) $

Now, substitute $ f(-2) = -8 $ into $ g(x) $:

\[ g(-8) = 2(-8) + 1 = -16 + 1 = -15 \]