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

Jul 25, 2018

Domain: $\text{ all reals", " or in interval notation: } \left(- \infty , \infty\right)$

Range: $y \ge 3 , \text{ or in interval notation: } \left[3 , \infty\right)$

#### Explanation:

Given: $f \left(x\right) = {x}^{2} - 4 x + 7 \text{ }$ 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 $A {x}^{2} + B x + C = 0$, the vertex can be found as $\left(- \frac{B}{2 A} , f \left(- \frac{B}{2 A}\right)\right)$

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

vertex: $\left(2 , 3\right)$ The lowest $y$-value is $3$

Range: $y \ge 3 , \text{ or in interval notation: } \left[3 , \infty\right)$

Graph of $f \left(x\right) = {x}^{2} - 4 x + 7$:

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: $\sqrt{x - 2}$:
$\text{ }$Domain: x >= 2; " Range: " y >= 0
graph{sqrt(x - 2) [-2, 5, -2, 5]}

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

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