Suppose that $g$ is the inverse of a function $f$. If $f ( 4 ) = 3$ and $f ^ { t } ( 4 ) = 2$, find $g ^ { t } ( 3 )$.
A) 3
B) $\frac { 1 } { 2 }$
C) 4
D) $\frac { 2 } { 3 }$

Question

Quizplus · Accepted Answer

Since $g$ is the inverse of $f$, we have $g ( f ( x ) ) = x$ for all $x$ in the domain of $f$. Using this property, we can write
$$2 = f ^ { t } ( 4 ) = f ( f ^ { t - 1 } ( 4 ) ) = f ( f ^ { t - 2 } ( 3 ) ) = \cdots = f ( 3 ) .$$
Therefore, we know that $g ( 3 ) = 4$. Using the same property again, we have
$$g ^ { t } ( 3 ) = f ( g ^ { t - 1 } ( 3 ) ) = f ( 4 ) = 3.$$
Thus, the answer is $\boxed{\textbf{(B) }\frac{1}{2}}.$

Suppose That ggg Is the Inverse of a Function fff If f(4)=3f ( 4 ) = 3f(4)=3

Suppose That $g$ Is the Inverse of a Function $f$ If $f ( 4 ) = 3$