# How do you graph, find any intercepts, domain and range of f(x)=(1/4)^(x+3)-2?

Apr 10, 2017

Doman is $\left(- \infty , \infty\right)$, range is $\left(- 2 , \infty\right)$, $y$-intercept is $- 1 \frac{63}{64}$ and $x$-intercept is $- \frac{7}{2}$.

#### Explanation:

$f \left(x\right) = {\left(\frac{1}{4}\right)}^{x + 3} - 2$

$= {\left({4}^{- 1}\right)}^{x + 3} - 2$

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

$= {2}^{- 2 x - 6} - 2$

As any power of $2$ is possible, there are no limitations on value of $x$ and doman is $\left(- \infty , \infty\right)$

However, as $x \to \infty$, as ${2}^{- \infty} \to 0$, value of $f \left(x\right)$ cannot go below $- 2$ and range is $\left(- 2 , \infty\right)$

Further when $x = 0$, $f \left(x\right) = {2}^{- 6} - 2 = \frac{1}{64} - 2 = - 1 \frac{63}{64}$ and hence $y$-intercept is $- 1 \frac{63}{64}$ and as $f \left(x\right) = 0$ is at ${2}^{- 2 x - 6} = 2$ or $- 2 x - 6 = 1$ i.e. $x = - \frac{7}{2}$, $x$-intercept is $- \frac{7}{2}$.

The graph appears as follows.

graph{(1/4)^(x+3)-2 [-10, 10, -5, 5]}