To determine whether $f(x) = -4x + 8$ and $g(x) = \frac{x + 10}{-4}$ are inverse functions, we need to use function composition. Specifically, we calculate:

1. $(f \circ g)(x)$, which means substituting $g(x)$ into $f(x)$.
2. $(g \circ f)(x)$, which means substituting $f(x)$ into $g(x)$.

Step 1: Compute $(f \circ g)(x)$

Substitute $g(x) = \frac{x + 10}{-4}$ into $f(x)$: $f(g(x)) = f\left(\frac{x + 10}{-4}\right) = -4\left(\frac{x + 10}{-4}\right) + 8.$ Simplify: $f(g(x)) = (-4) \cdot \frac{x + 10}{-4} + 8 = (x + 10) + 8 = x + 18.$

Step 2: Compute $(g \circ f)(x)$

Substitute $f(x) = -4x + 8$ into $g(x)$: $g(f(x)) = g(-4x + 8) = \frac{(-4x + 8) + 10}{-4}.$ Simplify: $g(f(x)) = \frac{-4x + 18}{-4} = x - \frac{18}{-4} = x - \frac{18}{-4}.$

To verify whether f(x) = -4x + 8 } andLet's continue the process more cleanly. The question is whether \(f(x) = -4x + 8 and $g(x) = \frac{x + 10}{-4}$ are inverses.

We already began the process of composition:

Step 1: Compute $f(g(x))$

Substituting $g(x) = \frac{x + 10}{-4}$ into $f(x) = -4x + 8$: $f(g(x)) = f\left(\frac{x + 10}{-4}\right) = -4\left(\frac{x + 10}{-4}\right) + 8.$ Simplify: $f(g(x)) = x + 10 + 8 = x + 18.$ Thus, $f(g(x)) = x + 18$, which is not equal to $x$.

Step 2: Compute $g(f(x))$

Substitute $f(x) = -4x + 8$ into $g(x) = \frac{x + 10}{-4}$: $g(f(x)) = g(-4x + 8) = \frac{(-4x + 8) + 10}{-4}.$ Simplify: $g(f(x)) = \frac{-4x + 18}{-4} = x - \frac{18}{-4}.$ Simplify further: $g(f(x)) = x + 4.5.$ Thus, $g(f(x)) = x + 4.5$, which is not equal to $x$.

Conclusion:

Since $f(g(x)) \neq x$ and $g(f(x)) \neq x$, the functions $f(x)$ and $g(x)$ are not inverses.

Would you like further clarification?

Here are some related questions to expand understanding:

1. How do you formally prove two functions are inverses?
2. Can you find the inverse of $f(x) = -4x + 8$ directly?
3. What properties must a function have to possess an inverse?
4. Why does function composition reveal whether functions are inverses?
5. How does graphing help visualize inverse functions?

Tip: Always verify both $f(g(x)) = x$ and $g(f(x)) = x$ when checking for inverse relationships!