Find \(f\left(\frac1x\right)+f(x)\)
Question
If
\[ f(x)=\frac{x-1}{x+1} \]then
\[ f\left(\frac1x\right)+f(x) \]is equal to __________.
Solution
Given
\[ f(x)=\frac{x-1}{x+1} \]First find
\[ f\left(\frac1x\right) \] \[ f\left(\frac1x\right) = \frac{\frac1x-1}{\frac1x+1} \]Simplify numerator and denominator:
\[ = \frac{\frac{1-x}{x}}{\frac{1+x}{x}} \] \[ = \frac{1-x}{1+x} \] \[ = -\frac{x-1}{x+1} \]Therefore,
\[ f\left(\frac1x\right)=-f(x) \]Hence,
\[ f\left(\frac1x\right)+f(x) \] \[ =-f(x)+f(x) \] \[ =0 \]