Question

Consider the function $f\left(x\right)=\frac{x}{x-1}.$ Which of the following statements are correct ?

• A. $f$ has the same domain and range
• B. $f$ has its own inverse
• C. $f$ is not injective
• D. $f$ is neither odd nor even

Answer 1

Answer: (A), (B), (D)

$f$ has its own inverse
$f$ has the same domain and range
$f$ is neither odd nor even

$f\left(x\right)=\frac{x}{x-1}$
Therefore Domain is $(-\infty ,1)\bigcup (1,\infty )$
Now the inverse for the given domain will be
$y(x-1)=x$
$xy-y=x$
$xy-x=y$
$x(y-1)=y$
$x=\frac{y}{y-1}$
Hence range is $(-\infty ,1)\bigcup (1,\infty )$
Hence $f\left(x\right)$ is the inverse of itself.
Also $f\left(x\right)+f(-x)\ne 0$ and $f\left(x\right)\ne f(-x)$
Thus $f\left(x\right)$ is not an even or odd function.
Hence, options 'A', 'B' and 'D' are correct.