Question

Find f(g(x)) and g (f(x)) and determine whether each pair of functions ƒ and g are inverses of each other. f(x) = 4x + 9 and g(x) = (x-9)/4

Accepted Answer

First, find the composition $$ f(g(x)) . $$This means you substitute $$ g(x) $$ into $$ f(x) . So$$, replace every $$ x  in$$ $$ f(x) = 4x + 9 $$ with $$ g(x) = \frac{x - 9}{4} . $$The expression becomes $$ f\left(g(x)\right) = 4 \left( \frac{x - 9}{4} \right) + 9 . $$Next, simplify the expression for $$ f(g(x)) . $$Multiply 4 by $$ \frac{x - 9}{4} $$ and then add 9. This will help you see if the composition simplifies to $$ x . $$Now, find the composition $$ g(f(x)) . $$This means you substitute $$ f(x) $$ into $$ g(x) . $$Replace every $$ x  in$$ $$ g(x) = \frac{x - 9}{4} $$ with $$ f(x) = 4x + 9 . $$The expression becomes $$ g\left(f(x)\right) = \frac{(4x + 9) - 9}{4} . $$Simplify the expression for $$ g(f(x)) . $$Subtract 9 from $$ 4x + 9 $$ and then divide by 4. Check if this simplifies to $$ x . $$Finally, determine if $$ f $$ and $$ g $$ are inverses by checking if both compositions $$ f(g(x)) $$ and $$ g(f(x)) $$ simplify to $$ x . If $$both do, then $$ f $$ and $$ g $$ are inverse functions of each other.

Find f(g(x)) and g (f(x)) and determine whether each pair of functions ƒ and g are inverses of each other. f(x) = 4x + 9 and g(x) = (x-9)/4

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Key Concepts

Function Composition

Inverse Functions

Algebraic Manipulation