# Question #59f7b

Jan 2, 2018

See the explanation.

#### Explanation:

If a function $f \left(x\right)$ has an inverse, it must be bijective, or one-to-one onto.

The figure is the example of a bijective function. If you choose one element from the right elipse, you can know the corresponding element in the left oval.

It is a good way to draw a graph of the function if you want to know whether it is bijective.

$f \left(x\right) = 2 x + 1$ is bijective.
graph{2x+1 [-5, 5, -5, 5]}

$f \left(x\right) = {x}^{2}$ is not bijective.
graph{x^2 [-5, 5, -5, 5]}

Then, how about $f \left(x\right) = x \tan \left(\frac{\pi x}{2}\right)$?
graph{xtan((pix)/2) [-1, 1, -5, 5]}

The function $f \left(x\right) = x \tan \left(\frac{\pi x}{2}\right)$ $\left(- 1 < x < 1\right)$ is not bijective.
For example, $f \left(\frac{1}{2}\right) = f \left(- \frac{1}{2}\right) = \frac{1}{2}$.
Therefore, it has no inverse.