Answer
a) Domain: $(-\infty,\infty)$
Range: $(2,\infty)$
b) $y=\log_3 (x-2)$
Domain: $(2,\infty)$
Range: $(\infty,\infty)$
Work Step by Step
We are given the function:
$y=3^x+2$
a) The domain of the function is the set of all real numbers:
$(-\infty,\infty)$
$3^x>0$
$3^x+2>0+2$
$y>2$
The range is:
$(2,\infty)$
b) Determine the inverse:
$y=3^x+2$
$y-2=3^x$
Interchange $x$ and $y$:
$x-2=3^y$
$\log_3 (x-2)=y$
The inverse is:
$y=\log_3 (x-2)$
The domain of the inverse is the range of the given function:
$(2,\infty)$
The range of the inverse is the domain of the given function:
$(-\infty,\infty)$
