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The function of x is given as f(x) and we have to find the inverse of the function of x as f -1(x).
Let us consider a function f(x) is given and we have to find out the inverse of it.
We can say that y = f(x).
The inverse of the function can be defined as x = g(y), where f(y) is the inverse function.
Here g(y) = f -1(y).
By replacing y with x in g(y) we can get f -1(x).
Solution:
Given equation:
\[f(x)=\frac{7^x-7^{-x}}{2}\]
Let, y = f(x):
\[\Rightarrow y=\frac{7^x-7^{-x}}{2}\]
\[\Rightarrow y=\frac{7^{2x}-1}{2\cdot7^x}\]
\[\Rightarrow 2\cdot7^x\cdot y=7^{2x}-1\]
\[\Rightarrow 7^{2x}-2\cdot7^x\cdot y-1=0\]
\[\Rightarrow (7^x)^2-2y(7^x)-1=0\]
It is a quadratic equation;
\[\Rightarrow 7^x=\frac{2y\pm\sqrt{4y^2+4}}{2}\]
\[\Rightarrow 7^x=y\pm\sqrt{y^2+1}\]
LHS will always be positive hence RHS must be positive,
We can see that:
\[\sqrt{y^2+1}>y\]
So we have to neglect negative term,
\[\Rightarrow 7^x=y+\sqrt{y^2+1}\]
\[\Rightarrow x=\log_7\left(y+\sqrt{y^2+1}\right)\]
Question related to Algebra, Function.
