what is the domain and range of f(x)={ |x-1| , 0<x<4 and 4 , x<0?

Question

how to get the domain and range of a function

Answer 1

If you think of a function like a machine that takes an input and produces an output, then the domain is "everything that you're allowed to put into the machine that won't break it" and the range is "every output that the machine could possibly produce".

So you have have a piecewise function:

f(x) = { 4, x < 0
         { |x - 1|, 0 < x < 4

Since you didn't use ≤, your function doesn't seem to have an output for f(0). That is, we know what happens for all x < 0 and for all x between 0 and 4, noninclusive, but we don't know what happens at x = 0, so it's not part of the domain. Likewise, the function doesn't specify any output for x ≥ 4, so as you've specified it, the domain is:

x < 0 or 0 < x < 4

or using interval notation:

x ∈ (-∞, 0) U (0, 4)

The range is all possible values we can get OUT of the function.

For all negative x, the only output is 4, so that's definitely part of the range.

For x values between 0 and 1, x - 1 is negative, and so | x - 1 | would be graphed as a line segment with slope of -1, going from (0, 1) to (1, 0) but not including (1, 0). So 0 ≤ f(x) < 1 is definitely part of the range as well.

For x values between 1 and 4, x - 1 is positive, and so | x - 1 | would be graphed as a line segment with slope of 1, going from (1, 0) to (4, 3) but not including (4, 3). So 0 ≤ f(x) < 3 is also part of the range.

Now the range of the function is the "union" of the ranges for each section.

{4} U [0, 1) U [0, 3) = [0, 3) U {4}

That is, the range is 4 and real numbers between 0 and 3, including 0 but not 3.

MsMath's answer should not include x = 3 as a possibility, the way you have specified the problem.

Answer 2

Domain just represents the x values.
Domain = {x | 0 < x < 4 and x < 0}
The largest |x-1| could be is when x = 4
|4-1| = |3| = 3
The smallest it could be is 0 (because absolute value is never negative.
So |x-1| is always between 0 and 3 when 0 < x < 4
f(x) is always 4 when x < 0
Range = { y | 0 <= y < 3, y = 4}

Yes, I forgot about not including the 3 (since x = 4 is not included)