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.