# How do you graph, find the intercepts and state the domain and range of f(x)=6^x+3?

May 30, 2017

y-intercept $\to \left(x , y\right) = \left(0 , 4\right)$

No x-intercept

domain $\textcolor{w h i t e}{.} \to \left\{x : x \in \left(- \infty , + \infty\right)\right\}$

range $\text{ } \to \left\{y : y \in \left(+ 3 , + \infty\right)\right\}$

#### Explanation:

Set $\text{ } y = {6}^{x} + 3$

$\textcolor{b l u e}{\text{Determine the y-intercept}}$

Set $x = 0$ giving

$y = {6}^{0} + 3 = 4$
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$\textcolor{b l u e}{\text{Determine the x-intercept}}$

Set $y = 0 = {6}^{x} + 3$

${6}^{x} = - 3$

Take logs of both sides

$x \ln \left(6\right) = \ln \left(- 3\right)$

But ln(-3) is undefined thus there is no x-intercept
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$\textcolor{b l u e}{\text{Determine end behaviour } x \to - \infty}$

${\lim}_{x \to - \infty} {6}^{x} + 3 = {\lim}_{z \to + \infty} \frac{1}{6} ^ z + 3 \to k = 3$
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$\textcolor{b l u e}{\text{Determine end behaviour } x \to + \infty}$

${\lim}_{x \to + \infty} {6}^{x} + 3 \text{ " ->" } k + 3 = \infty + 3 = \infty$
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