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Mathematics Question on Functions

Which of the following functions is inverse of itself?

A

f(t)=(1−t)(1+t)f\left(t\right) = \frac{\left(1-t\right)}{\left(1+t\right)}

B

f(t)=(1−t2)(1+t2)f\left(t\right) = \frac{\left(1-t^{2}\right)}{\left(1+t^{2}\right)}

C

f(t)=4log tf\left(t\right) = 4^{log\, t}

D

f(t)=2tf(t) = 2^t

Answer

f(t)=(1−t)(1+t)f\left(t\right) = \frac{\left(1-t\right)}{\left(1+t\right)}

Explanation

Solution

Let y=f(t)y=f (t)
∴t=f−1(y)\therefore t=f^{-1}(y)
Now, y=f(t)=1−t1+ty=f (t) =\frac{1-t}{1+t}
⇒y+ty=1−t\Rightarrow y+ty=1-t
⇒t+ty=1−y\Rightarrow t+ty=1-y
⇒t=1−y1+y\Rightarrow t=\frac{1-y}{1+y}
i.e., f−1(y)=1−y1+yf^{-1}(y) =\frac{1-y}{1+y}
or f−1(t)=1−t1+tf^{-1}(t)=\frac{1-t}{1+t}
Thus, this function is inverse of itself