Find the range of $f(x)=(-1)^{\lfloor x \rfloor} (x-{\lfloor x \rfloor})$

Question

Find the range of $f(x)=(-1)^{\lfloor x \rfloor} (x-{\lfloor x \rfloor})$

I don't know how to work with this function, and how to find the range. I tried to find the domain of its inverse, but i can't do it either.

Any hints?

Answer 1

Note that $x-\lfloor x\rfloor$ is the fractional part of $x$, hence is between zero and one. Because $\lfloor x\rfloor$ can be even or odd, the range is $(-1,1)$.

Answer 2

Here's my shot

x - [x] = {x} where {x} € [0,1) Also, [x] is always an integer.

Now for the given function we can note that ( loosely writing ) f(x) = (-1)^(integer) · {x} Or f(x) = +/- {x} depending on the value of x ( [x] being odd or even )

So, the range of f(x) = (-1,1)

This can also be used to modify the graph of fr(x) to graph f(x) (taking x-axis reflection where [x] is odd)

Answer 3

$$f(x)=(-1)^{\lfloor x\rfloor}\{x\},\ 0\le \{x\}\lt 1$$ Since the fractional part is being multiplied by a factor of $(-1)^{\lfloor x\rfloor}$ it would just introduce the set of numbers created by multiplying by $-1$ to the well known range of the fractional part function, thereby making the range $(-1,1)$.