Question(b) $f(x)=3 x$ $\begin{array}{l} g(x)=3 x \\ f(g(x))= \\ g(f(x))= \end{array}$ $f$ and $g$ are inverses of each other $f$ and $g$ are not inverses of each other

Studdy Solution

STEP 1

1. The functions $f(x) = 3x$ and $g(x) = 3x$ are both linear functions.
2. To determine if $f$ and $g$ are inverses, we need to check if $f(g(x)) = x$ and $g(f(x)) = x$ .
3. If both compositions equal $x$ , then $f$ and $g$ are inverses of each other.

STEP 2

1. Calculate $f(g(x))$ .
2. Calculate $g(f(x))$ .
3. Determine if $f$ and $g$ are inverses by checking if both compositions equal $x$ .

STEP 3

Calculate $f(g(x))$ :
Given: $g(x) = 3x$
Substitute $g(x)$ into $f(x)$ : $f(g(x)) = f(3x) = 3(3x) = 9x$

STEP 4

Calculate $g(f(x))$ :
Given: $f(x) = 3x$
Substitute $f(x)$ into $g(x)$ : $g(f(x)) = g(3x) = 3(3x) = 9x$

STEP 5

Determine if $f$ and $g$ are inverses:
For $f$ and $g$ to be inverses, both $f(g(x))$ and $g(f(x))$ must equal $x$ .
From previous steps: $f(g(x)) = 9x \neq x$ $g(f(x)) = 9x \neq x$
Since neither composition equals $x$ , $f$ and $g$ are not inverses of each other.
The conclusion is that $f$ and $g$ are not inverses of each other.

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