Question

Let `f(x) = ([x]+1)/({x}+1) ` for `f: [0, (5)/(2) ) to ((1)/(2) , 3]`, where `[*]` represents the greatest integer function and `{*}` represents the fractional part of x.
Draw the graph of `y= f(x)`. Prove that `y=f(x)` is bijective. Also find the range of the function.

Answer 1

We have `f(x) = ([x]+1)/({x}+1)`
`" "={{:((1)/(x+1)",",, 0le x lt 1), ((2)/(x)",",,1le xlt 2), ((3)/(x-1)",",,2le x lt (5)/(2)):}`
Each branch of the function is part of a rectangular hyperbola.
The graph of the function is as shown in the function is as shown in the following figure.

From the figure, the range of the function is `[1//2, 3]`.
Clearly, `f(x)` is a bijective function.