https://file.aiquickdraw.com/mathbot/file/1729138955235ph3ypzhx.jpg

What is the doman and range and state the asymptote

The given function is:

\[
h(x) = e^{-x} - 5
\]

### Domain:
The domain of the function \( h(x) = e^{-x} - 5 \) refers to the set of all possible input values (values of \( x \)) for which the function is defined. Since \( e^{-x} \) is defined for all real numbers, the domain of this function is:

\[
\text{Domain} = (-\infty, \infty)
\]

### Range:
The range is the set of all possible output values (values of \( h(x) \)). Since the exponential function \( e^{-x} \) always produces positive values (but never reaches 0), we know:

- \( e^{-x} > 0 \) for all real \( x \),
- Therefore, \( e^{-x} - 5 > -5 \).

Thus, the range of the function is:

\[
\text{Range} = (-5, \infty)
\]

### Horizontal Asymptote:
As \( x \to \infty \), the value of \( e^{-x} \) approaches 0. Therefore, \( h(x) = e^{-x} - 5 \) approaches:

\[
h(x) \to 0 - 5 = -5 \quad \text{as} \quad x \to \infty.
\]

Thus, the horizontal asymptote is:

\[
y = -5
\]

Would you like further details on graphing this function?

Here are 5 related questions:
1. What would happen to the graph if the function was \( e^{x} - 5 \) instead of \( e^{-x} - 5 \)?
2. How does shifting the graph vertically by 5 units affect the domain and range?
3. What is the behavior of the function as \( x \to -\infty \)?
4. How would you compute the derivative of \( h(x) \) and what does it tell us about the function's slope?
5. Can you find the x-intercept of the function?

**Tip**: When identifying horizontal asymptotes, remember to focus on the behavior of the function as \( x \to \infty \) or \( x \to -\infty \).

What is the domain and range and state the asymptote for the function h(x) = e^(-x) - 5?

Find the domain, range, and horizontal asymptote of the function h(x) = e^(-x) - 5. Learn how exponential functions behave and how to determine their key features. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation