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

1 Answer
Oct 17, 2016

Answer:

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]}