Question

How do you find the domain and range of `f(x)=(x-1)^2 +2`?

Answer 1

The domain is all real numbers `RR` or in interval notation `(-oo,oo)`.

The range is `y>=2` or in interval notation `[2,oo)`.

Explanation:

`f(x)=(x-1)^2+2=color(blue)1(x-color(red)1)^2+color(red)2`

The easiest way to find the domain and range of a quadratic function is to look at the graph.

The general equation of a parabola in vertex form is
`y=a(x-h)^2+k` where `(h,k)` is the vertex.

A positive `a` means the parabola is upward facing (U shaped) and a negative `a` means it is downward facing (an upside down U shape).

The vertex of this example is then `(color(red)1,color(red)2)`.
`a=+color(blue)1` so the shape is an upward facing parabola.

The domain is found by considering all the possible values of `x`.
Looking at the graph, you can see that `x` goes all the way from negative infinity to positive infinity. The domain can be expressed as all real numbers `RR`, or in interval notation, `(-oo,oo)`.

The range is found by considering all the possible values of `y`.
There are no values of `y` below `y=2`, so `y>=2`. In interval notation, the range is `[2,oo)`.

graph{(x-1)^2+2 [-10.12, 9.88, -2.4, 7.6]}