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Question

# Consider the function$f ( x ) = - \frac { 1 } { 2 } x + 5$Use composition to show that f(x) and the equation you wrote are inverses.

Solution

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for inverse

\begin{align*} &f\left( f^{-1}\left( x\right)\right)=x\\ \\ &f\left( -2\cdot{x}+10\right)\\ \\ &=-\frac{1}{2}\left(-2\cdot{x}+10 \right)+5\\ \\ &=x-5+5\\ \\ &=x \end{align*}

$\textbf{so, }$ $f^{-1}\left( x\right)$ $\textbf{ is the inverse of }$ $f\left( x\right)$

for inverse

\begin{align*} &f^{-1}\left( f\left( x\right)\right)=x\\ \\ &f^{-1}\left( -\dfrac{1}{2}x+5\right)+10\\ \\ &=x-10+10\\ \\ &=x \end{align*}

$\textbf{so, }$ $f^{-1}\left( x\right)$ $\textbf{ is the inverse of }$ $f\left( x\right)$

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