Question

5. Determine whether or not the following functions from real numbers to real numbers are bijections. If they are bijections, then find the inverse. If they are not bijections, then explain why not.

(a) f(x) = [2x]

(b) f(x) = −7x

(c) f(x) = 7x 3 – 5

(d) f(x) = x 2 − 5

Answer 1

For a function to be a bijection, it should be one-one as well as onto

For finding out the inverse, we first find x in terms of y and the replace y with x and x with y

a)

means greatest integer function

So, here for many values of x we get the same value of y, hence the function is not a bijection

b)  

The function is always decreasing which means for one value of x we get one value of y, this is one-one, we also know that the range of this function is has range . So, the function is also onto

Hence, the function is a bijection

Finding inverse

• Find x in terms of y
• Replace y with x and then x with y  

So, is the inverse of

c)  

The function is always increasing which means for one value of x we get one value of y, this is one-one, we also know that the range of this function is has range . So, the function is also onto

Hence, the function is a bijection

Finding inverse

• Find x in terms of y
• Replace y with x and then x with y  

So, is the inverse of

d)  

This function is first decreasing till and then increasing after . This means the function can take a value at more than one value of x, so this is not one-one

Hence, the function is not a bijection