Question

use transformations of the graph of f(x)=3^x 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 + 5

graph g(x)=3^x + 5 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

the equation of the asymptote for g(x)=3^x + 5 is \boxed{}. (type an equation.)

the domain of g(x)=3^x + 5 is \boxed{}.
(type your answer in interval notation.)

the range of g(x)=3^x + 5 is \boxed{}.
(type your answer in interval notation.)

Explanation:

Step1: Analyze Asymptote Transformation

For the parent function \( f(x) = 3^x \), the horizontal asymptote is \( y = 0 \). When we have \( g(x) = 3^x + 5 \), this is a vertical shift up by 5 units. So the asymptote of \( g(x) \) is \( y = 5 \).

Step2: Determine Domain of \( g(x) \)

The function \( 3^x \) is defined for all real numbers \( x \). Adding 5 (a vertical shift) does not affect the domain. So the domain of \( g(x) = 3^x + 5 \) is all real numbers, which in interval notation is \( (-\infty, \infty) \).

Step3: Determine Range of \( g(x) \)

The range of \( f(x) = 3^x \) is \( (0, \infty) \) because \( 3^x > 0 \) for all \( x \). For \( g(x) = 3^x + 5 \), we add 5 to each output of \( f(x) \). So \( 3^x + 5 > 0 + 5 = 5 \). Thus, the range is \( (5, \infty) \).

Answer:

The equation of the asymptote for \( g(x) = 3^x + 5 \) is \( y = 5 \).
The domain of \( g(x) = 3^x + 5 \) is \( (-\infty, \infty) \).
The range of \( g(x) = 3^x + 5 \) is \( (5, \infty) \).