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

1 Answer
Jul 25, 2018

Answer:

Domain: #" all reals", " or in interval notation: "(-oo, oo)#

Range: # y >= 3, " or in interval notation: "[3, oo)#

Explanation:

Given: #f(x) = x^2 - 4x + 7" "# A quadratic function

Unless a function is limited, the domain is all reals.

Quadratic functions which are graphs of parabolas have a maximum or minimum, the vertex. This vertex determines the range values.

If the equation is in the from #Ax^2 + Bx + C = 0#, the vertex can be found as #(-B/(2A), f(-B/(2A)))#

#-B/(2A) = 4/2 = 2; " "f(2) = 2^2 -4*2 + 7 = 3#

vertex: #(2, 3)# The lowest #y#-value is #3#

Range: # y >= 3, " or in interval notation: "[3, oo)#

Graph of #f(x) = x^2 - 4x + 7#:

graph{x^2 - 4x + 7 [-5, 5, -2, 10]}

Here are some examples of functions that are limited in domain and range:

  1. Contains a square root: #sqrt(x-2)#:
    #" "#Domain: #x >= 2; " Range: " y >= 0#
    graph{sqrt(x - 2) [-2, 5, -2, 5]}

  2. Rational functions: #x/(x+4):" "# contain asymptotes
    #" "#Domain: #x != -4; " Range: " y != 1#
    graph{x/(x+4) [-15, 5, -10, 10]}

  3. Exponential functions: #2^x#:
    #" Domain: all reals; Range: " y>0#
    graph{2^x [-5, 10, -10, 30]}