Math Snap

STEP 1

1. We have a piecewise-defined function $f(x)$.
2. The function has two cases:
- $f(x) = 7^{-x-4} - 3$ for $x < -4$
- $f(x) = \frac{-36}{x+8} + 7$ for $x > -4$
3. We need to evaluate $f(x)$ at $x = -6$.

STEP 2

1. Determine which case of the piecewise function applies for $x = -6$.
2. Substitute $x = -6$ into the appropriate expression.
3. Simplify the expression to find the exact value of $f(-6)$.

STEP 3

Determine which case applies for $x = -6$.
Since $-6 < -4$, we use the first case of the piecewise function:
$$f(x) = 7^{-x-4} - 3$$

STEP 4

Substitute $x = -6$ into the expression $7^{-x-4} - 3$.
$$f(-6) = 7^{-(-6)-4} - 3$$

STEP 5

Simplify the exponent:
$$f(-6) = 7^{6-4} - 3$$ $$f(-6) = 7^{2} - 3$$

Calculate $7^2$ and then subtract 3:
$$f(-6) = 49 - 3$$ $$f(-6) = 46$$ The exact value of $f(-6)$ is:
$$\boxed{46}$$