Let's solve each part of the question step by step. We have two functions defined as follows:

$f(x) = -5x + 3$ $g(x) = \frac{1}{5}(x - 3)$

(a) Find $f(g(x))$

To find $f(g(x))$, we substitute $g(x)$ into $f(x)$: $f(g(x)) = f\left(\frac{1}{5}(x - 3)\right)$ Now, substitute this expression into $f(x) = -5x + 3$: $f\left(\frac{1}{5}(x - 3)\right) = -5\left(\frac{1}{5}(x - 3)\right) + 3$ Simplify the expression: $= -(x - 3) + 3$ $= -x + 3 + 3$ $= -x + 6$

So, $f(g(x)) = -x + 6$.

(b) Find $g(f(x))$

To find $g(f(x))$, we substitute $f(x)$ into $g(x)$: $g(f(x)) = g(-5x + 3)$ Now, substitute this expression into $g(x) = \frac{1}{5}(x - 3)$: $g(-5x + 3) = \frac{1}{5}((-5x + 3) - 3)$ Simplify the expression inside the parentheses: $= \frac{1}{5}(-5x)$ $= -x$

So, $g(f(x)) = -x$.

(c) Determine whether the functions $f$ and $g$ are inverses of each other

Two functions $f$ and $g$ are inverses of each other if $f(g(x)) = x$ and $g(f(x)) = x$.

From parts (a) and (b), we found: $f(g(x)) = -x + 6$ $g(f(x)) = -x$

Since neither $f(g(x)) = x$ nor $g(f(x)) = x$, the functions $f$ and $g$ are not inverses of each other.

Would you like more details or have questions about any steps?

Further Questions

1. How do we find the inverse of a function in general?
2. What properties do inverse functions satisfy?
3. Can you explain the steps in simplifying $f(g(x))$?
4. What is the difference between composition of functions and inverse functions?
5. How can we verify if a function is one-to-one?

Tip

When testing if two functions are inverses, remember to check both $f(g(x)) = x$ and $g(f(x)) = x$; both conditions must be true for them to be inverses.