Determine Algebraically Whether F and G Are Inverse Functions $f ( x ) = \sqrt { x + 6 }$

question 30

Multiple Choice

Determine algebraically whether f and g are inverse functions. $f ( x ) = \sqrt { x + 6 }$ g(x) = x² - 6,x ? 0

• A)Yes,f and g are inverse functions. \(f ( g ( x ) ) = f \left( x ^ { 2 } - 6 \right) = \sqrt { \left( x ^ { 2 } - 6 \right) + 6 } = \sqrt { x ^ { 2 } } = x\) \(g ( f ( x ) ) = g ( \sqrt { x + 6 } ) = ( \sqrt { x + 6 } ) ^ { 2 } - 6 = x + 6 - 6 = x\)
• B)No,f and g are not inverse functions. \(f ( g ( x ) ) = f \left( x ^ { 2 } - 6 \right) = \sqrt { \left( x ^ { 2 } - 6 \right) + 6 } = \sqrt { x ^ { 2 } } = x\) \(g ( f ( x ) ) = g ( \sqrt { x + 6 } ) = ( \sqrt { x + 6 } ) ^ { 2 } - 6 = - x + 6 - 6 = - x\)

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