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To analyze the function \( f(x) = 4^x \), let’s go through the details step-by-step.

### Part a: Domain and Range

1. **Domain**: The function \( f(x) = 4^x \) is defined for all real values of \( x \) because any real number \( x \) can be used as an exponent for 4. Thus, the domain is:
   \[
   (-\infty, \infty)
   \]

2. **Range**: Since \( 4^x \) is an exponential function with a positive base greater than 1, it will always yield positive values. As \( x \to -\infty \), \( 4^x \to 0 \), but it never actually reaches 0. As \( x \to \infty \), \( 4^x \to \infty \). Therefore, the range is:
   \[
   (0, \infty)
   \]

### Part b: Asymptote

Since \( f(x) = 4^x \) approaches 0 as \( x \to -\infty \) but never actually reaches it, there is a horizontal asymptote at \( y = 0 \).

### Final Answers

- **Domain**: \((-\infty, \infty)\)
- **Range**: \((0, \infty)\)
- **Asymptote**: \( y = 0 \)

Would you like further details on exponential functions or how to determine asymptotes?

### Additional Questions

1. What changes would occur to the domain and range if the function were \( f(x) = 4^{-x} \)?
2. How does the base of the exponential function (in this case, 4) affect the graph's growth rate?
3. What would happen to the asymptote if a constant were added to \( f(x) = 4^x \)?
4. How does the graph of \( f(x) = 4^x \) compare to \( f(x) = e^x \)?
5. If the function were \( f(x) = 4^{x+2} \), how would the graph shift?

### Tip

For exponential functions of the form \( f(x) = a^x \), where \( a > 1 \), the graph always increases as \( x \) increases and approaches 0 as \( x \) decreases. This pattern is consistent regardless of the base (as long as \( a > 1 \)).

Analyze the function f(x) = 4^x. Part a: Enter the domain and range in interval notation. Part b: Enter the asymptote.

Learn how to analyze the exponential function f(x) = 4^x by finding its domain, range, and asymptote. This problem covers key concepts in exponential functions and properties of asymptotes, suitable for students in Grades 9-12.

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