https://file.aiquickdraw.com/mathbot/file/1730807042815itjvzqt0.jpg

To solve this problem, let's analyze the provided graph.

### Part (a): Finding the Exponential Function \( f(x) \)
The graph appears to represent an exponential function, typically in the form:
\[
f(x) = a \cdot b^x
\]
where \( a \) is the initial value (y-intercept) and \( b \) is the base of the exponential.

From the graph:
- The point at \( (0, 1) \) suggests that when \( x = 0 \), \( f(x) = 1 \). This implies that \( a = 1 \), since \( f(0) = a \cdot b^0 = a \).
- Another point on the graph, possibly at \( (1, 2) \), indicates that \( f(1) = 2 \). Using this point, we can solve for \( b \):
  \[
  f(1) = a \cdot b = 2
  \]
  Since \( a = 1 \), we have \( b = 2 \).

Thus, the function is:
\[
f(x) = 2^x
\]

### Part (b): Domain and Range
For the function \( f(x) = 2^x \):
- **Domain**: All real numbers, \( (-\infty, \infty) \), because exponential functions are defined for all real \( x \)-values.
- **Range**: \( (0, \infty) \), because \( 2^x \) is always positive but never zero.

Would you like further details on any step?

#### Related Questions
1. What would the function be if one of the points was \( (1, 3) \) instead of \( (1, 2) \)?
2. How do we determine the y-intercept of an exponential function from a graph?
3. Can the base \( b \) in an exponential function be negative? Why or why not?
4. How would the graph change if the base \( b \) was less than 1 (like \( 0.5 \))?
5. What are some real-world applications of exponential functions?

**Tip:** For exponential functions, knowing just two points can often determine the entire function if they include the y-intercept.

Use the given points and curve to identify the equation for the EXPONENTIAL function graphed below. (a) f(x) = ? (b) Find the Domain and the Range.

Learn how to identify the equation for an exponential function based on a graph and determine its domain and range. This problem involves analyzing the graph to determine the base and initial value of the function, applying exponential function properties. Suitable for students in Grades 9-12.

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