Question

Graph $g\left(x\right)=3^{x-1}+5$ using transformations. Determine the domain, the range, and horizontal asymptote of $g$.

Short answer

The graph of the function:

Step-by-step solution

Step 1. Given information.

The function to be plotted is:

$g\left(x\right)=3^{x-1}+5$

Step 2. The graph of y=3x.

$y=3^x$

When $x=0$, then

x	y	(x, y)
0	1	(0,1)
1	3	(1,3)
-1	$\frac{1}{3}$	localid="1647756706009" $\left(-1,\frac{1}{3}\right)$
-3	$\frac{1}{27}$	localid="1647756712423" $\left(-3,\frac{1}{27}\right)$

Locate these points in the coordinate plane and match all points then extend the graph.

Step 3. To draw graph of y=3x-1

Horizontally shift the above graph by 1 unit towards right.

Step 4. To graph g(x)=3x-1+5

Vertically shift the above graph by 5 units upwards.

From this graph we can see that as $x\to -\infty$

$g\left(x\right)\to 5$

So, the horizontal asymptote of the function is:

$y=5$.

Step 5. To find the domain and the range of the function.

From the above graph we can see that values of $x$is $\left(-\infty ,\infty \right)$.

The value of the $y$is $\left(5,\infty \right)$.

So the domain is :$\left(-\infty ,\infty \right)$.

The range is:$\left(5,\infty \right)$.