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Help with functions

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Define $g$ by $g(x)=5x-4$. If $g(x)=f^{-1}(x)-3$ and $f^{-1}(x)$ is the inverse of the function $f(x)=ax+b$, find $5a+5b$.

Jul 20, 2018

#1
+22884
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Define $g$ by $g(x)=5x-4$. If $g(x)=f^{-1}(x)-3$ and $f^{-1}(x)$ is the inverse of the function $f(x)=ax+b$,

find $5a+5b$.

$$\begin{array}{|rcl|rcl|} \hline g(x)&=& 5x-4 & f(x) &=& ax + b \quad & | \quad x \leftrightarrow y \\ g(x)&=& f^{-1}(x)-3 & x &=& ay + b \\ && & ay &=& x-b \\ f^{-1}(x)-3 &=& 5x -4 & y &=& \dfrac{x-b}{a} \\ f^{-1}(x) &=& 5x -4+3 & f^{-1}(x) &=& \dfrac{x-b}{a} \\ \hline f^{-1}(x) &=& {\color{red}5}x {\color{green}-1} & f^{-1}(x) &=& {\color{red}\dfrac1a} x {\color{green}-\dfrac ba} \\ \hline \end{array}$$

compare:

$$\begin{array}{|rcll|} \hline 5 &=& \dfrac1a \\\\ \mathbf{a} &\mathbf{=} & \mathbf{\dfrac15} \\ \hline \end{array} \begin{array}{|rcll|} \hline -1 &=& -\dfrac ba \\\\ 1 &=& \dfrac ba \\\\ b &=& a \\ \mathbf{b} &\mathbf{=} & \mathbf{\dfrac15} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline 5a+5b &=& 5\cdot \dfrac15 + 5\cdot \dfrac15 \\ &=& 1+1 \\ &=& 2 \\ \hline \end{array}$$

Jul 20, 2018