Question

exponential functions
equation
$y = 6^{x - 2}+\frac{1}{2}$
domain: $(-\infty,\infty)$
range: $(\frac{1}{2},\infty)$
x - intercept: none
y - intercept: $(0.5,0)$
increasing: $(-\infty,\infty)$
decreasing: none
absolute max: none
absolute min: none
end behavior:
as $x\to -\infty$, $y\to\\_(\frac{1}{2},0)$_
as $x\to\infty$, $y\to\\_\infty_$
asymptote:
$y = \frac{1}{2}$

Explanation:

Step1: Analyze end - behavior as $x\to-\infty$

For the exponential function $y = 6^{x - 2}+\frac{1}{2}$, when $x\to-\infty$, the term $6^{x - 2}=\frac{6^{x}}{6^{2}}$. Since $6>1$, as $x\to-\infty$, $6^{x}\to0$. So $y\to\frac{1}{2}$.

Step2: Analyze end - behavior as $x\to\infty$

When $x\to\infty$, the term $6^{x - 2}$ grows without bound because the base $a = 6>1$. So $y\to\infty$.

Answer:

As $x\to-\infty$, $y\to\frac{1}{2}$; as $x\to\infty$, $y\to\infty$