Question

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

Answer 1

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]}