Question

Use the definition of inverses to determine whether ƒ and g are inverses. f(x) = 2/(x+6), g(x) = (6x+2)/x

Accepted Answer

Recall that two functions $$ f $$ and $$ g $$ are inverses if and only if $$ f(g(x)) = x $$ and $$ g(f(x)) = x $$ for all $$ x  in $$the domains of the compositions. Start by finding the composition $$ f(g(x)) . $$Substitute $$ g(x) = \frac{6x + 2}{x} $$ into $$ f(x) = \frac{2}{x} + 6 $$, so $$ f(g(x)) = \frac{2}{\frac{6x + 2}{x}} + 6 . $$Simplify the expression $$ f(g(x)) = \frac{2}{\frac{6x + 2}{x}} + 6  by $$multiplying numerator and denominator appropriately to eliminate the complex fraction. Next, find the composition $$ g(f(x)) . $$Substitute $$ f(x) = \frac{2}{x} + 6 $$ into $$ g(x) = \frac{6x + 2}{x} $$, so $$ g(f(x)) = \frac{6\left(\frac{2}{x} + 6\right) + 2}{\frac{2}{x} + 6} . $$Simplify $$ g(f(x))  by $$distributing and combining like terms in numerator and denominator, then check if both $$ f(g(x)) $$ and $$ g(f(x)) $$ simplify to $$ x . If $$both do, then $$ f $$ and $$ g $$ are inverses.

Use the definition of inverses to determine whether ƒ and g are inverses. f(x) = 2/(x+6), g(x) = (6x+2)/x

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Key Concepts

Definition of Inverse Functions

Function Composition

Simplifying Rational Expressions