To solve this problem, let's analyze the given information and use it to determine the value of $b$.

Given:

• The function $f(x) = a^x + b$, where $a$ and $b$ are constants.
• $f(x)$ has an x-intercept at $(2, 0)$:
  • This means that when $x = 2$, $f(2) = 0$.
• $f(x)$ has a y-intercept at $(0, -323)$:
  • This means that when $x = 0$, $f(0) = -323$.

Step-by-Step Solution:

1. Use the y-intercept condition:

   • At $x = 0$: $f(0) = a^0 + b = -323$
   • Since $a^0 = 1$: $1 + b = -323$ $b = -323 - 1 = -324$

2. Verify using the x-intercept condition:

   • At $x = 2$: $f(2) = a^2 + b = 0$
   • Substitute $b = -324$: $a^2 - 324 = 0$ $a^2 = 324$ $a = \pm 18$

Final Answer:

$\boxed{b = -324}$

Would you like a further breakdown of the calculations or the reasoning? Here are some related questions for further practice:

1. If $f(x) = a^x + b$ and $a = 2$, find the x-intercept when $b = -8$.
2. Determine the y-intercept of the function $f(x) = 5^x - 10$.
3. How does changing the value of $b$ affect the graph of $f(x) = a^x + b$?
4. Find the derivative of $f(x) = a^x + b$ with respect to $x$.
5. If the x-intercept of $f(x) = a^x + b$ is at $(3, 0)$, what is the relationship between $a$ and $b$?

Tip: When finding intercepts, remember that the x-intercept occurs where $f(x) = 0$ and the y-intercept occurs where $x = 0$.