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For each pair of functions [tex]f[/tex] and [tex]g[/tex] below, find [tex]f(g(x))[/tex] and [tex]g(f(x))[/tex]. Then, determine whether [tex]f[/tex] and [tex]g[/tex] are inverses of each other.

Simplify your answers as much as possible.
(Assume that your expressions are defined for all [tex]x[/tex] in the domain of the composition. You do not have to indicate the domain.)

(a) [tex]f(x) = x + 3[/tex]
(b) [tex]f(x) = -\frac{1}{4x}, x \neq 0[/tex]

1. [tex]g(x) = x - 3[/tex]
\begin{align}
f(g(x)) &= \square \\
g(f(x)) &= \square
\end{align}
[tex]f[/tex] and [tex]g[/tex] are inverses of each other.

2. [tex]g(x) = \frac{1}{4x}, x \neq 0[/tex]
\begin{align}
f(g(x)) &= \square \\
g(f(x)) &= \square
\end{align}
[tex]f[/tex] and [tex]g[/tex] are not inverses of each other.



Answer :

GinnyAnswer
Let's analyze each pair of functions step by step to determine the compositions [tex]\( f(g(x)) \)[/tex] and [tex]\( g(f(x)) \)[/tex], and then check whether [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are inverses.

## Pair (a)
Given functions:
[tex]\[ f(x) = x + 3 \][/tex]
[tex]\[ g(x) = x - 3 \][/tex]

### Step 1: Calculate [tex]\( f(g(x)) \)[/tex]
[tex]\[ g(x) = x - 3 \][/tex]
Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(g(x)) = f(x - 3) = (x - 3) + 3 = x \][/tex]

### Step 2: Calculate [tex]\( g(f(x)) \)[/tex]
[tex]\[ f(x) = x + 3 \][/tex]
Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(f(x)) = g(x + 3) = (x + 3) - 3 = x \][/tex]

Since both compositions [tex]\( f(g(x)) = x \)[/tex] and [tex]\( g(f(x)) = x \)[/tex], [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are indeed inverses of each other.

## Pair (b)
Given functions:
[tex]\[ f(x) = -\frac{1}{4x}, x \neq 0 \][/tex]
[tex]\[ g(x) = \frac{1}{4x}, x \neq 0 \][/tex]

### Step 1: Calculate [tex]\( f(g(x)) \)[/tex]
[tex]\[ g(x) = \frac{1}{4x} \][/tex]
Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(g(x)) = f\left(\frac{1}{4x}\right) = -\frac{1}{4 \left(\frac{1}{4x}\right)} = -\frac{1}{\frac{1}{x}} = -x \][/tex]

### Step 2: Calculate [tex]\( g(f(x)) \)[/tex]
[tex]\[ f(x) = -\frac{1}{4x} \][/tex]
Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(f(x)) = g\left(-\frac{1}{4x}\right) = \frac{1}{4 \left( -\frac{1}{4x} \right)} = \frac{1}{-\frac{1}{x}} = -x \][/tex]

Since neither composition [tex]\( f(g(x)) = -x \)[/tex] nor [tex]\( g(f(x)) = -x \)[/tex] simplifies to [tex]\( x \)[/tex], [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are not inverses of each other.

## Summary:
(a) For [tex]\( f(x) = x + 3 \)[/tex] and [tex]\( g(x) = x - 3 \)[/tex]:
[tex]\[ f(g(x)) = x \][/tex]
[tex]\[ g(f(x)) = x \][/tex]
[tex]\( f \)[/tex] and [tex]\( g \)[/tex] are inverses of each other.

(b) For [tex]\( f(x) = -\frac{1}{4x}, x \neq 0 \)[/tex] and [tex]\( g(x) = \frac{1}{4x}, x \neq 0 \)[/tex]:
[tex]\[ f(g(x)) = -x \][/tex]
[tex]\[ g(f(x)) = -x \][/tex]
[tex]\( f \)[/tex] and [tex]\( g \)[/tex] are not inverses of each other.

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