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# Functions

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If f(x) = a/(x + 2), solve for the value of a such that $f(0) = f^{-1}(-3a)$.

Jul 30, 2021

#1
+26213
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If

$$f(x) = \dfrac{a}{x + 2}$$,

solve for the value of $$a$$ such that $$f(0) = f^{-1}(-3a)$$.

My attempt:
$$\begin{array}{|rcll|} \hline f(x) &=& \dfrac{a}{x + 2} \\\\ f(0) &=& \dfrac{a}{0 + 2} \\\\ \mathbf{f(0)} &=& \mathbf{\dfrac{a}{2}} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline f(x) &=& \dfrac{a}{x+2} \\\\ x &=& \dfrac{a}{f^{-1}(x)+2} \\\\ x\Big(f^{-1}(x)+2\Big) &=& a \\\\ xf^{-1}(x)+2x &=& a \\\\ xf^{-1}(x) &=& a-2x \\\\ f^{-1}(x) &=& \dfrac{a-2x}{x} \quad | \quad x =-3a\\\\ f^{-1}(-3a) &=& \dfrac{a-2(-3a)}{-3a} \\\\ f^{-1}(-3a) &=& \dfrac{a+6a}{-3a} \\\\ f^{-1}(-3a) &=& \dfrac{7a}{-3a} \\\\ \mathbf{f^{-1}(-3a)} &=& \mathbf{-\dfrac{7}{3}} \\\\ \hline f(0) &=& f^{-1}(-3a) \\\\ \dfrac{a}{2} &=& -\dfrac{7}{3} \\\\ \mathbf{a} &=& \mathbf{-\dfrac{14}{3}} \\ \hline \end{array}$$

Jul 31, 2021