Questions: For each pair of functions (f) and (g) below, find (f(g(x))) and (g(f(x))). Then, determine whether (f) and (g) are inverses of each other. Simplify your answers as much as possible. (Assume that your expressions are defined for all (x) in the domain of the composition. You do not have to indicate the domain.) (a) (f(x)=-1/(6x), x neq 0) (g(x)=-1/(6x), x neq 0) (f(g(x))=) (g(f(x))=) (f) and (g) are inverses of each other (f) and (g) are not inverses of each other (b) (f(x)=x+4) (g(x)=x+4) (f(g(x))=) (g(f(x))=) (f) and (g) are inverses of each other

Solution Steps

To determine if two functions \( f \) and \( g \) are inverses of each other, we need to check if \( f(g(x)) = x \) and \( g(f(x)) = x \). We will compute these compositions for each pair of functions and check if they simplify to \( x \).

• \( f(x) = -\frac{1}{6x} \)
• \( g(x) = -\frac{1}{6x} \)

For these functions, compute \( f(g(x)) \) and \( g(f(x)) \).

• \( f(x) = x + 4 \)
• \( g(x) = x + 4 \)

For these functions, compute \( f(g(x)) \) and \( g(f(x)) \).

Given:

• \( f(x) = -\frac{1}{6x} \)
• \( g(x) = -\frac{1}{6x} \)

Compute:

• \( f(g(x)) = f\left(-\frac{1}{6x}\right) = -\frac{1}{6\left(-\frac{1}{6x}\right)} = x \)
• \( g(f(x)) = g\left(-\frac{1}{6x}\right) = -\frac{1}{6\left(-\frac{1}{6x}\right)} = x \)

Since both compositions simplify to \( x \), \( f \) and \( g \) are inverses of each other.

Given:

• \( f(x) = x + 4 \)
• \( g(x) = x + 4 \)

Compute:

• \( f(g(x)) = f(x + 4) = (x + 4) + 4 = x + 8 \)
• \( g(f(x)) = g(x + 4) = (x + 4) + 4 = x + 8 \)

Since neither composition simplifies to \( x \), \( f \) and \( g \) are not inverses of each other.

Final Answer