Use transformations of the graph of ​f(x)equals3 Superscript x to graph the given function. Be sure to graph and give the equation of the asymptote. Use the graph to determine the​ function's domain and range. If​ applicable, use a graphing utility to confirm your​ hand-drawn graphs.
​g(x)equals3 Superscript x Baseline plus 4
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Part 1
Graph ​g(x)equals3 Superscript x Baseline plus 4 and its asymptote. Use the graphing tool to graph the function as a solid curve and the asymptote as a dashed line.

Part 2
The equation of the asymptote for ​g(x)equals3 Superscript x Baseline plus 4 is 
  
enter your response here. ​(Type an​ equation.)
Part 3
The domain of ​g(x)equals3 Superscript x Baseline plus 4 is 
  
enter your response here.
​(Type your answer in interval​ notation.)
Part 4
The range of ​g(x)equals3 Superscript x Baseline plus 4 is 
  
enter your response here.
​(Type your answer in interval​ notation.

To solve the problem of graphing \( g(x) = 3^x + 4 \), we need to follow several steps. Let's break it down:

### Part 1: Graph \( g(x) = 3^x + 4 \) and its Asymptote
1. **Start with the base function \( f(x) = 3^x \)**:
   - This function is an exponential curve that passes through (0, 1) and rises rapidly.
   - The horizontal asymptote for \( f(x) = 3^x \) is \( y = 0 \).

2. **Transform the graph of \( f(x) = 3^x \) to get \( g(x) = 3^x + 4 \)**:
   - This transformation is a vertical shift upward by 4 units.

3. **Graphing the function**:
   - Plot key points of \( 3^x \) and then shift them up by 4 units.
   - For example, at \( x = 0 \), \( 3^0 = 1 \), so \( g(0) = 1 + 4 = 5 \).

4. **The asymptote**:
   - The horizontal asymptote of \( g(x) = 3^x + 4 \) is \( y = 4 \).

### Part 2: Equation of the Asymptote
The equation of the asymptote for \( g(x) = 3^x + 4 \) is:
\[ y = 4 \]

### Part 3: Domain of \( g(x) = 3^x + 4 \)
The domain of the function \( g(x) = 3^x + 4 \) is all real numbers because exponential functions are defined for all \( x \):
\[ \text{Domain: } (-\infty, \infty) \]

### Part 4: Range of \( g(x) = 3^x + 4 \)
Since \( 3^x \) is always positive and approaches 0 as \( x \) goes to negative infinity, \( g(x) = 3^x + 4 \) will be always greater than 4:
\[ \text{Range: } (4, \infty) \]

### Graph:
Here is a sketch of the graph of \( g(x) = 3^x + 4 \):

- Draw the curve of \( 3^x \) and shift it 4 units up.
- The curve should pass through points such as (0, 5), (1, 7), (-1, \approx 4.33), etc.
- The horizontal asymptote is a dashed line at \( y = 4 \).

Would you like any further details or explanations on this graph? Here are some additional questions you might have:

1. How do you find the asymptote of an exponential function?
2. Can you explain how transformations affect the graph of a function?
3. What are some real-world applications of exponential functions?
4. How do you solve exponential equations analytically?
5. Can you demonstrate the inverse function of an exponential function?

**Tip:** When graphing functions, always identify the transformations applied to the base function to correctly plot the graph and determine the domain, range, and asymptotes.

Learn how to graph the function g(x) = 3^x + 4, including identifying its domain, range, and asymptote. Understand the effect of vertical shifts on exponential functions. Suitable for high school students studying functions and transformations.

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