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)=5x-9 and g(x) = (x+5)/9

Accepted Answer

First, find the composition $$ f(g(x))  by $$substituting $$ g(x) = \frac{x+5}{9} $$ into $$ f(x) = 5x - 9 . $$This means replacing every $$ x  in$$ $$ f(x) $$ with $$ \frac{x+5}{9} . So$$, write $$ f\left(g(x)\right) = 5 \left( \frac{x+5}{9} \right) - 9 . $$Next, simplify the expression for $$ f(g(x))  by $$distributing the 5 and combining like terms carefully. This will give you a simplified function in terms of $$ x . $$Then, find the composition $$ g(f(x))  by $$substituting $$ f(x) = 5x - 9 $$ into $$ g(x) = \frac{x+5}{9} . $$Replace every $$ x  in$$ $$ g(x) $$ with $$ 5x - 9 $$, so write $$ g\left(f(x)\right) = \frac{(5x - 9) + 5}{9} . $$Simplify the expression for $$ g(f(x))  by $$combining like terms in the numerator and then dividing by 9 to get a simplified function in terms of $$ x . $$Finally, determine whether $$ f $$ and $$ g $$ are inverses by checking if both compositions $$ f(g(x)) $$ and $$ g(f(x)) $$ simplify to $$ x . If $$both equal $$ x $$, 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)=5x-9 and g(x) = (x+5)/9

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Key Concepts

Function Composition

Inverse Functions

Algebraic Manipulation