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

1 Answer
Dec 27, 2017

Answer:

#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]}