Question: (b) \( f(x) = -\frac{3}{x}, \, x \neq 0 \) \( g(x) = \frac{3}{x}, \, x \neq 0 \) \(…

(b) \( f(x) = -\frac{3}{x}, \, x \neq 0 \)

\( g(x) = \frac{3}{x}, \, x \neq 0 \)

\( f(g(x)) = \boxed{\quad} \)

\( g(f(x)) = \boxed{\quad} \)

\( \bigcirc f \text{ and } g \text{ are inverses of each other} \)

\( \bigcirc f \text{ and } g \text{ are not inverses of each other} \)

Solution

Step 1 Compute \( f(g(x)) \). \[ f(g(x)) = f\left( \frac{3}{x} \right) = -\frac{3}{\frac{3}{x}} = -x \] Step 2 Compute \( g(f(x)) \). \[ g(f(x)) = g\left( -\frac{3}{x} \right) = \frac{3}{-\frac{3}{x}} = -x \] Conclusion: Since \( f(g(x)) \neq x \) and \( g(f(x)) \neq x \), ○ f and g are not inverses of each other.