Question

What is the domain and range of the function `g(x)=sqrt(x-1)`?

Answer 1

Hello,

• The domain of `g` is `[1,+infty[`,
• The range of `g` is `[0,+infty[`.

Indeed,

• A real number `x` is in the domain `D` if and only if `sqrt(x-1)` exists, it means `x-1 >= 0`, or `x>=1`. Therefore `D = [1,+oo[`.

• The range is the set `V` of all the values of the function `g` :

1) Because `g(x)` is a square root, `g(x) >= 0` : therefore, `R subset [0,+oo[`.

2) On the other hand, if `y>= 0`, you can write `y = g(x)` if you consider `x= y^2+1`. Therefore `[0,+oo[ subset R` and finally `R= [0,+oo[`.

Graphically :

Domain is the projection of the curve of `g` on x-axes
Range is the projection of the curve of `g` on y-axes

graph{sqrt(x-1) [-1.75, 18.25, -1.88, 8.12]}