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Let's break down the function \( f(x) = 20^x \) and determine the properties as requested.

### a. Domain:
The domain of an exponential function like \( f(x) = 20^x \) is all real numbers because there are no restrictions on the input \( x \).
- **Domain**: \( (-\infty, \infty) \)

### b. Range:
For an exponential function of the form \( f(x) = a^x \) (where \( a > 0 \)), the function never touches 0 and only takes positive values.
- **Range**: \( (0, \infty) \)

### c. End Behavior:
- As \( x \to -\infty \), the function approaches 0 because \( 20^x \) gets smaller as \( x \) decreases.
- As \( x \to \infty \), the function grows without bound because \( 20^x \) increases rapidly.
  - \( \lim_{x \to -\infty} f(x) = 0 \)
  - \( \lim_{x \to \infty} f(x) = \infty \)

So the end behavior is:
- **As \( x \to -\infty \), \( f(x) \to 0 \)**
- **As \( x \to \infty \), \( f(x) \to \infty \)**

### d. x-intercept(s):
To find the x-intercept, set \( f(x) = 0 \) and solve for \( x \):
\[
20^x = 0
\]
However, since \( 20^x > 0 \) for all real \( x \), there is no solution.
- **x-intercept(s)**: **DNE**

### e. y-intercept(s):
The y-intercept occurs when \( x = 0 \). Substituting \( x = 0 \) into the function:
\[
f(0) = 20^0 = 1
\]
- **y-intercept(s)**: \( (0, 1) \)

### Summary:
- **a. Domain**: \( (-\infty, \infty) \)
- **b. Range**: \( (0, \infty) \)
- **c. End Behavior**:
  - As \( x \to -\infty \), \( f(x) \to 0 \)
  - As \( x \to \infty \), \( f(x) \to \infty \)
- **d. x-intercept(s)**: DNE
- **e. y-intercept(s)**: \( (0, 1) \)

Would you like further details on any of these? Here are some related questions you can explore:

1. How do transformations affect the domain and range of an exponential function?
2. What happens to the intercepts of an exponential function with a base smaller than 1?
3. How can you shift the graph of \( f(x) = 20^x \) to create a horizontal asymptote other than \( y = 0 \)?
4. What is the difference in the growth rate of exponential functions with different bases (e.g., \( 20^x \) vs. \( 2^x \))?
5. How do you solve for x in more complex exponential equations?

**Tip**: The behavior of exponential functions is highly sensitive to their base, affecting their growth rate and how quickly they approach asymptotes.

Determine each of the following properties of the exponential function f(x) = 20^x.

Explore the domain, range, intercepts, and end behavior of the exponential function f(x) = 20^x. This guide covers essential concepts for high school students in Grades 9-12.

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