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.