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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 \( x = 0 \).

The function f is defined by f(x) = a^x + b, where a and b are constants. In the xy-plane, the graph of y = f(x) has an x-intercept at (2, 0) and a y-intercept at (0, -323). What is the value of b?

Learn how to solve for the constant b in the exponential function f(x) = a^x + b, given the x-intercept at (2, 0) and the y-intercept at (0, -323). This problem involves key algebraic concepts like intercepts and the behavior of exponential functions, suitable for students in Grades 10-12.

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