The function $g(x) = 2^x + 1$ is an exponential function. To sketch its graph, we can follow these steps:

Key Characteristics of $g(x)$:

1. Base Function: The base function is $f(x) = 2^x$, which is an exponential function with a base of 2. The graph of $f(x) = 2^x$ grows rapidly as $x$ increases.
2. Vertical Shift: The "+1" outside of the exponential term shifts the graph of $2^x$ vertically upward by 1 unit.

Steps for Sketching:

1. Identify the y-intercept: When $x = 0$, $g(0) = 2^0 + 1 = 1 + 1 = 2$ So, the graph passes through the point $(0, 2)$.

2. End behavior:

   • As $x \to \infty$, $2^x \to \infty$, so $g(x) \to \infty$.
   • As $x \to -\infty$, $2^x \to 0$, so $g(x) \to 1$. Therefore, the graph approaches the horizontal asymptote $y = 1$ from above but never touches it.

3. Plot a few points:

   • $g(1) = 2^1 + 1 = 2 + 1 = 3$ → Point $(1, 3)$
   • $g(-1) = 2^{-1} + 1 = \frac{1}{2} + 1 = 1.5$ → Point $(-1, 1.5)$
   • $g(2) = 2^2 + 1 = 4 + 1 = 5$ → Point $(2, 5)$

4. Asymptote: The graph has a horizontal asymptote at $y = 1$.

General Behavior:

• The graph starts close to $y = 1$ as $x \to -\infty$.
• It passes through the points $(0, 2)$, $(1, 3)$, and grows exponentially as $x \to \infty$.

Would you like me to generate the graph of this function for you?

Relative Questions:

1. What happens to the graph of $g(x) = 2^x + 1$ if you change the base to another value, such as 3?
2. How does adding or subtracting a constant inside the exponent $g(x) = 2^{x+c}$ affect the graph?
3. Can you explain how the graph of $g(x) = 2^x$ would look without the vertical shift?
4. How would the graph of $g(x) = -2^x + 1$ change compared to the current function?
5. What is the effect of multiplying the entire function by a constant, such as $g(x) = 3(2^x + 1)$?

Tip: When graphing exponential functions, always check for horizontal asymptotes and make sure to calculate key points to guide your sketch.