# What is the domain and range of h(x)=6 - 4^x?

Mar 18, 2016

Domain: $\left(- \infty . \infty\right)$
Range: $\left(- \infty , 6\right)$

#### Explanation:

The domain of a function is the range of real numbers the variable X can take such that $h \left(x\right)$ is real. The range is the set of all values which $h \left(x\right)$ can take when $x$ is assigned a value in the domain.

Here we have a polynomial involving the subtraction of an exponential. The variable is really only involved in the $- {4}^{x}$ term, so we'll work with that.

There are three primary values to check here: $x < - a , x = 0 , x > a$, where $a$ is some real number. ${4}^{0}$ is simply 1, so $0$ is in the domain. Plugging in various positive and negative integers, one determines that ${4}^{x}$ yields a real result for any such integer. Thus, our domain is all real numbers, here represented by $\left[- \infty , \infty\right]$

How about the range? Well, first note the range of the second part of the expression, ${4}^{x}$. If one puts in a large positive value, one gets a large positive output; putting in 0 yields 1; and putting in a 'large' negative value yields a value very close to 0. Thus, the range of ${4}^{x}$ is $\left(0 , \infty\right)$. If we place these values into our initial equation, we learn that the lower bound is $- \infty$ ($6 - {4}^{x}$ goes to $- \infty$ as x goes to $\infty$), and the upper bound is 6 (h(x)) goes to $6$ as $x \to - \infty$)

Thus, we arrive at the following conclusions.
Domain: $\left(- \infty , \infty\right)$
Range: $\left(- \infty , 6\right)$