Question

How do you find the domain and range of `f(x)=x^2-4x+7`?

Answer 1

`x inRR`
`y inRR,y>=3`

Explanation:

`f(x)" is defined for all "x inRR`

`rArr"domain is "x inRR`

`"to find the range we require to find the vertex"`

`"the equation of a parabola in "color(blue)"vertex form"` is.

`color(red)(bar(ul(|color(white)(2/2)color(black)(y=a(x-h)^2+k)color(white)(2/2)|)))`

`"where "(h,k)" are the coordinates of the vertex and a "`
`"is a multiplier"`

`• " if "a>0" then vertex is a minimum "uuu`

`• " if "a<0" then vertex is a maximum"nnn`

`"to obtain vertex form use "color(blue)"completing the square"`

`• " coefficient of the "x^2" term must be 1 which it is"`

`• " add/subtract "(1/2"coefficient of x-term")^2" to"`
`x^2-4x`

`rArry=x^2+2(-2)xcolor(red)(+4)color(red)(-4)+7`

`rArry=(x-2)^2+3`

`rArr" vertex "=(2,3)" and "a>0`

`rArr"range is "y inRR,y>=3`
graph{x^2-4x+7 [-10, 10, -5, 5]}