Question

watch the video and then solve the problem given below.
click here to watch the video.
begin by graphing f(x) = 3^x. then use transformations of this graph to graph the given function. be sure to graph and give the equation of the asymptote. use the graph to determine the functions domain and range. if applicable, use a graphing utility to confirm your hand - drawn graphs.
g(x)=3^x + 1
graph g(x)=3^x + 1 and its asymptote. use the graphing tool to graph the function as a solid curve and the asymptote as a dashed line.
click to enlarge graph

Explanation:

Step1: Analyze the parent function

The parent function is \( f(x) = 3^x \). The graph of \( y = 3^x \) has a horizontal asymptote at \( y = 0 \), domain \( (-\infty, \infty) \), and range \( (0, \infty) \).

Step2: Determine the transformation

For \( g(x) = 3^x + 1 \), this is a vertical shift of the parent function \( f(x) = 3^x \) upward by 1 unit.

Step3: Find the asymptote

A vertical shift upward by 1 unit means the horizontal asymptote of \( g(x) \) is also shifted upward by 1 unit. So the equation of the asymptote for \( g(x) \) is \( y = 1 \).

Step4: Determine domain and range

The domain of an exponential function of the form \( a^x + k \) (where \( a>0, a
eq1 \)) is always \( (-\infty, \infty) \) because we can plug in any real number for \( x \). For the range, since \( 3^x>0 \) for all real \( x \), then \( 3^x + 1>0 + 1 = 1 \). So the range of \( g(x) \) is \( (1, \infty) \).

Step5: Graphing (conceptual)

To graph \( g(x) = 3^x + 1 \), start with the graph of \( f(x) = 3^x \) (which passes through \( (0, 1) \), \( (1, 3) \), \( (-1, \frac{1}{3}) \), etc.), then shift each point upward by 1 unit. For example, the point \( (0, 1) \) on \( f(x) \) becomes \( (0, 2) \) on \( g(x) \), \( (1, 3) \) becomes \( (1, 4) \), \( (-1, \frac{1}{3}) \) becomes \( (-1, \frac{1}{3}+1=\frac{4}{3}) \), and draw the curve. The asymptote \( y = 1 \) is a dashed horizontal line.

Answer:

• Asymptote equation: \( y = 1 \)
• Domain: \( (-\infty, \infty) \)
• Range: \( (1, \infty) \)
• Graph: The graph of \( g(x)=3^x + 1 \) is the graph of \( f(x)=3^x \) shifted up 1 unit, with a dashed horizontal asymptote at \( y = 1 \).