# What is the range of f(x) = -3^x + 4?

May 6, 2015

Write $y = - {3}^{x} + 4$

$\implies {3}^{x} = 4 - y$

Take $\ln$ of both sides

$\implies \ln {3}^{x} = \ln \left(4 - y\right)$

$\implies x = \ln \frac{4 - y}{\ln} 3$

Now notice that $\left(4 - y\right)$ cannot be negative nor zero!

$\implies 4 - y > 0 \implies y < 4$

Hence the range of $f \left(x\right)$ is $f \left(x\right) < 4$