The given problem asks to graph the exponential function $g(x) = 4^x + 2$, plot two points, draw the asymptote, and provide the domain and range of the function in interval notation.

Step-by-Step Solution:

1. Exponential Function: The given function is $g(x) = 4^x + 2$. This is an exponential function where the base is 4 and there is a vertical shift upwards by 2.

2. Plotting Two Points: To plot the graph, we need two points. Let's calculate these by substituting different values of $x$ into the function:

   • For $x = 0$: $g(0) = 4^0 + 2 = 1 + 2 = 3$ So, the point is $(0, 3)$.

   • For $x = 1$: $g(1) = 4^1 + 2 = 4 + 2 = 6$ So, the point is $(1, 6)$.

3. Asymptote: The function $g(x) = 4^x + 2$ has a horizontal asymptote. Since the graph of $4^x$ approaches zero as $x \to -\infty$, the horizontal asymptote is shifted up by 2 units. Thus, the horizontal asymptote is: $y = 2$

4. Domain: The exponential function is defined for all real values of $x$, so the domain is: $(-\infty, \infty)$

5. Range: Since $4^x$ is always positive and we add 2 to it, the smallest value of the function occurs when $x \to -\infty$, where $4^x$ approaches zero. Therefore, the range is: $(2, \infty)$

Summary:

• Points: $(0, 3)$, $(1, 6)$
• Asymptote: $y = 2$
• Domain: $(-\infty, \infty)$
• Range: $(2, \infty)$

Would you like more details on graphing or any of the steps?

Follow-Up Questions:

1. What happens to the graph as $x \to -\infty$?
2. How does the base of the exponential function affect the graph's growth rate?
3. Can you explain the effect of the constant "2" in the function $4^x + 2$?
4. How would the graph change if the function was $g(x) = 4^x - 2$?
5. What is the general form of an exponential function?

Tip: Always identify the asymptote in exponential functions to better understand their long-term behavior.