Practicing Success
If f: R – {–1} → R – {1} be a function defined by f(x) = \(\frac{x-1}{x+1}\), then A. f is one-one but not onto B. f is onto but not one-one C. f is one-one and onto D. f-1(x) = \(\frac{x+1}{x-1}\) E. (fof)(x) = -\(\frac{1}{x}\); x ≠ 0, -1 Choose the correct answer from the options given below |
A, D, E only C, D only B, E only C, E only |
C, E only |
Let x, y be two numbers in set R - {-1} such that f(x) = f(y) $\frac{x-1}{x+1}=\frac{y-1}{y+1}$ ⇒ Simplifying the equation. $f(x)=\frac{x-1}{x+1}=\frac{(x+1)}{x+1}-\frac{2}{x+1}=1-\frac{2}{x+1}$ $f'(x)=\frac{2}{(x+1)^2}>0$ ∴ f is one-one $f(x) = 1-\frac{2}{x+1}∈R$ ...(i) ∴ f is onto also & $y=\frac{x-1}{x+1}$ $\frac{y+1}{y-1}=\frac{x}{-1}⇒f^{-1}(x)=\frac{x+1}{1-x}⇒\frac{\frac{x-1}{x+1}-1}{\frac{x-1}{x+1}+1}=\frac{-2}{2x}=-\frac{1}{x}$ |