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

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
Transcript text: 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.) \[ \text { (a) } \begin{array}{l} f(x)=-\frac{1}{6 x}, x \neq 0 \\ g(x)=-\frac{1}{6 x}, x \neq 0 \\ f(g(x))= \\ g(f(x))=\square \end{array} \] $f$ and $g$ are inverses of each other $f$ and $g$ are not inverses of each other (b) $f(x)=x+4$ \[ \begin{array}{l} g(x)=x+4 \\ f(g(x))=\square \\ g(f(x))=\square \end{array} \] $f$ and $g$ are inverses of each other
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Solution

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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 \( x \).

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

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

(b) Functions:
  • \( f(x) = x + 4 \)
  • \( g(x) = x + 4 \)

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

Step 1: Evaluate \( f(g(x)) \) and \( g(f(x)) \) for Part (a)

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.

Step 2: Evaluate \( f(g(x)) \) and \( g(f(x)) \) for Part (b)

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

  • For part (a), \( f \) and \( g \) are inverses of each other: \(\boxed{\text{Yes}}\)
  • For part (b), \( f \) and \( g \) are not inverses of each other: \(\boxed{\text{No}}\)
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