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1. Let $f(x) = Ax - 2B^2$ and $g(x) = Bx$, where $B \neq 0$. If $f(g(1)) = 0$, what is $A$ in terms of $B$?

2. Suppose that $f$ is a function and $f^{-1}$ is the inverse of $f$. If $f(1)=2$, $f(2) = 6$, and $f(3)=5$, then what is $f^{-1}(f^{-1}(6))$?

Guest Jul 16, 2018
 #1
avatar+20744 
+2

1. Let $f(x) = Ax - 2B^2$ and $g(x) = Bx$, where $B \neq 0$. If $f(g(1)) = 0$, what is $A$ in terms of $B$?

2. Suppose that $f$ is a function and $f^{-1}$ is the inverse of $f$. If $f(1)=2$, $f(2) = 6$, and $f(3)=5$, then what is $f^{-1}(f^{-1}(6))$?

 

 

\(\begin{array}{|lrcll|} \hline x = 1 : & g(x) &=& g(1) \\ & g(1) &=& B\cdot 1 \\ & g(1) &=& B \\\\ & f(g(1)) &=& f(B) \\ x=B: & f(B) &=& A\cdot B - 2B^2 \\ f(B) = 0: & A\cdot B - 2B^2 &=& 0 \\ & A\cdot B&=& 2B^2 \\ & A &=& \dfrac{2B^2}{B} \quad & | \quad B \ne 0~ ! \\ &\mathbf{ A }& \mathbf{=}& \mathbf{ 2B } \\ \hline \end{array}\)

 

laugh

heureka  Jul 17, 2018
 #2
avatar+20744 
+2

1. Let $f(x) = Ax - 2B^2$ and $g(x) = Bx$, where $B \neq 0$. If $f(g(1)) = 0$, what is $A$ in terms of $B$?
2. Suppose that $f$ is a function and $f^{-1}$ is the inverse of $f$. If $f(1)=2$, $f(2) = 6$, and $f(3)=5$, then what is $f^{-1}(f^{-1}(6))$?

 

\(\begin{array}{|r|r|r|r|} \hline x & f(x) \\ \hline 1 & 2 & f(1) = 2 & f^{-1}(2) = 1 \\ 2 & 6 & f(2) = 6 & f^{-1}(6) = 2 \\ 3 & 5 & f(3) = 5 & f^{-1}(5) = 3 \\ \hline f^{-1}(x) & x \\ \hline \end{array}\)

 

\(\begin{array}{|rcll|} \hline f^{-1}(6) &=& 2 \\ f^{-1}(f^{-1}(6)) &=& f^{-1}(2) \\ &=& 1 \\\\ \mathbf{f^{-1}(f^{-1}(6))}& \mathbf{=}& \mathbf{1} \\ \hline \end{array}\)

 

 

laugh

heureka  Jul 17, 2018

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