Math Snap

STEP 1

1. We have two functions: $f(x) = r x + s$ and $g(x) = \frac{1}{r}(x - s)$.
2. The composition of functions involves substituting one function into another.
3. We need to find the compositions $(f \circ g)(x)$ and $(g \circ f)(x)$.
4. We will then compare these compositions to check if they are equal.

STEP 2

1. Find the expression for $(f \circ g)(x)$.
2. Find the expression for $(g \circ f)(x)$.
3. Compare the expressions to determine if they are equal.

STEP 3

To find $(f \circ g)(x)$, substitute $g(x)$ into $f(x)$.
Given:
$$f(x) = r x + s$$ $$g(x) = \frac{1}{r}(x - s)$$ Substitute $g(x)$ into $f(x)$:
$$(f \circ g)(x) = f\left(g(x)\right) = f\left(\frac{1}{r}(x - s)\right)$$

Now, compute $f\left(\frac{1}{r}(x - s)\right)$:
$$f\left(\frac{1}{r}(x - s)\right) = r\left(\frac{1}{r}(x - s)\right) + s$$ Simplify the expression:
$$= (x - s) + s$$ $$= x$$ Thus, $(f \circ g)(x) = x$.
The expression for $(f \circ g)(x)$ is $\boxed{x}$.