# Why is this function not invertible?

Why is the following function NOT invertible?

$$y[n] = \frac 1 {x[n-1]^2}$$

A function is invertible if two distinct inputs give two distinct outputs. Is there a quick way to check for invertibility?

• Draw a graph. And if you can draw a horizontal line over two or more points in it, it is not invertible. And for the function in your example y[n]=1/(x[n-1])^2 = 1/(-x[n-1])^2. So it has the same value for any pair of negative/positive value. – Eugene Sh. Oct 21 '15 at 19:40
• any other ways ? I won't be able to draw on the exam – user65652 Oct 21 '15 at 19:43
• In the case of the example the function is clearly even. Even functions are non-invertible. – Eugene Sh. Oct 21 '15 at 19:45
• how can you tell it is even by just looking at it? – user65652 Oct 21 '15 at 19:50
• When you see x to the power of even number it will always lead to a same result for both x and -x. – Eugene Sh. Oct 21 '15 at 19:52

$$x[n-1] = 1$$ $$y[n] = \frac 1 {x[n-1]^2} = \frac 1 {(1)^2} = 1$$
$$x[n-1] = -1$$ $$y[n] = \frac 1 {x[n-1]^2} = \frac 1 {(-1)^2} = 1$$