Question

How do you find the domain and range of `y=2abs(x-3)+5`?

Answer 1

Domain is all real numbers or `(-oo,oo)` and range is `y>=5` or `[5,oo)`.

Explanation:

If you look at the graph of `y=absx`, you will see the vertex is located at `(0,0)` with a V shape pointing up.

The graph of `y=2abs(x-3)+5` has a vertex at `(3,5)`.
The number inside the absolute value symbol indicates the translation in x, but with the opposite sign. The number added or subtracted indicates the translation in y. The coefficient in front of the absolute value symbol gives the slope of the lines in the graph, but you don't need that to find domain and range.

Here is a graph of `y=2abs(x-3)+5`
graph{y=2abs(x-3)+5 [-7.19, 12.81, -0.08, 9.92]}

To find the domain, look at the possible x values of the graph. Note that x can go all the way to either negative or positive infinity. The domain can then be written as "all real numbers" or `RR` or in interval notation, `(-oo,oo)`.

To find range, look at the possible y values of the graph. Note that the function only has y values when y is 5 and above. So the range can be written as `y>=5` or in interval notation, `[5,oo)`.
The square bracket next to the 5 indicates that 5 is included in the range.

In interval notation, infinity symbols always get a parenthesis, not a square bracket.

To figure out domain and range for a standard
`y=aabs(x-b)+c` without using a graph:

domain is always all real numbers or `(-oo,oo)`

range for a positive value of `a` is `y>=c` or `[c,oo)`
and because a negative a flips the graph down,
range for a negative value of `a` is `y<=c` or `(-oo,c]`