Find \(f\left(\frac1x\right)\)
Question
If
\[ f(x)=\frac{x-1}{x+1} \]then
\[ f\left(\frac1x\right) \]is equal to __________.
Solution
Given
\[ f(x)=\frac{x-1}{x+1} \]Replace \(x\) by \(\frac1x\):
\[ f\left(\frac1x\right) = \frac{\frac1x-1}{\frac1x+1} \]Simplify numerator and denominator:
\[ = \frac{\frac{1-x}{x}}{\frac{1+x}{x}} \] \[ = \frac{1-x}{1+x} \]Multiply numerator and denominator by \(-1\):
\[ = -\frac{x-1}{x+1} \]