# How do I find the range of the function y=-2^x+2?

Apr 22, 2015

Here's my attempt:

$y < 2$ is the range.

$y = - {2}^{x} + 2$ Can also be written as

$2 - y = {2}^{x}$

and also,

$\ln \left(2 - y\right) = x \ln 2$

$\implies 2 - y > 0 \implies y < 2$

May 21, 2015

The range of $y = - {2}^{x} + 2$ is (-infty;2).

I'd start from a known fact that ${a}^{x} > 0$ for all $a > 0$ and $x \in \mathbb{R}$.
So:

${2}^{x} > 0$
$- {2}^{x} < 0$
$- {2}^{x} + 2 < 2$
$y < 2$
y in (-infty;2)