Question

How do you find the domain and range of `f(x)= x^2- 6x + 8`?

Answer 1

`x inRR,y in[-1,+oo)`

Explanation:

`f(x)" is defined for all real values of x"`

`rArr"domain is "x inRR`

`"to determine the range express "f(x)" in "color(blue)"vertex form"`

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

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

`"using the method of "color(blue)"completing the square"`

`f(x)=x^2+2(-3)xcolor(red)(+9)color(red)(-9)+8`

`color(white)(f(x))=(x-3)^2-1`

`rArrcolor(magenta)"vertex "=(3,-1)`

`"to determine if the vertex is a max/min then"`

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

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

`"here "a=1>0rArr" vertex is a minimum"`

`rArr"range is "[-1,+oo)`
graph{x^2-6x+8 [-10, 10, -5, 5]}