Questions: Let f(x)=3x-8 and g(x)=x+2. Find f(g(x)) and g(f(x)). f(g(x))= (Simplify your answer.)

Transcript text: Let $f(x)=3 x-8$ and $g(x)=x+2$. Find $f(g(x))$ and $g(f(x))$. \[ f(g(x))= \] $\square$ (Simplify your answer.)

Solution

Step 1: Find $ f(g(x)) $

To find $ f(g(x)) $, substitute $ g(x) $ into $ f(x) $: \[ f(g(x)) = f(x + 2) \]

Step 2: Substitute $ g(x) $ into $ f(x) $

Replace $ x $ in $ f(x) $ with $ g(x) = x + 2 $: \[ f(x + 2) = 3(x + 2) - 8 \]

Step 3: Simplify the expression

Expand and simplify the expression: \[ f(x + 2) = 3x + 6 - 8 = 3x - 2 \]

Step 4: Find $ g(f(x)) $

To find $ g(f(x)) $, substitute $ f(x) $ into $ g(x) $: \[ g(f(x)) = g(3x - 8) \]

Step 5: Substitute $ f(x) $ into $ g(x) $

Replace $ x $ in $ g(x) $ with $ f(x) = 3x - 8 $: \[ g(3x - 8) = (3x - 8) + 2 \]

Step 6: Simplify the expression

Combine like terms: \[ g(3x - 8) = 3x - 6 \]

\[ f(g(x)) = 3x - 2 \] \[ g(f(x)) = 3x - 6 \]

$ f(g(x)) = \boxed{3x - 2} $
$ g(f(x)) = \boxed{3x - 6} $

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