# How do you find the domain and range of f(x)=-x^2+4x-3?

Jun 26, 2017

$x \in \mathbb{R}$
$y \le 1$

#### Explanation:

This is a parabola, so the domain is all real numbers.

We can find the range by completing the square.

$f \left(x\right) = - {x}^{2} + 4 x - 3$

$f \left(x\right) = - {x}^{2} + 4 x - 4 + 1$

$f \left(x\right) = - \left({x}^{2} - 4 x + 4\right) + 1$

$f \left(x\right) = - {\left(x - 2\right)}^{2} + 1$

The number outside the square (1, in this case) tells us the endpoint of our range. The sign in front of the square determines which direction to shade on the number line.

In this case, the number is 1 and the sign is negative, so our range is everything less than or equal to 1.

$y \le 1$