Consider the graph of the exponential function f (x) = 32. Which of the follow The graph crosses the x-axis. The graph has a horizontal asymptote at y = 0. The graph crosses the y-axis. The graph has a vertical asymptote at x = 0. As x increases, y also increases. As x increases, y decreases.

Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.
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People keep getting it wrong. Is there only one answer? It is not the combination of “2,3,5”. I got it wrong. I am confused.
### Understanding the Exponential Function \( f(x) = 3^x \)

Consider the graph of the exponential function \( f(x) = 3^x \). Which of the following statements are correct? (Select all that apply.)

- [ ] The graph crosses the x-axis.
- [x] The graph has a horizontal asymptote at \( y = 0 \).
- [ ] The graph crosses the y-axis.
- [ ] The graph has a vertical asymptote at \( x = 0 \).
- [x] As \( x \) increases, \( y \) also increases.
- [ ] As \( x \) increases, \( y \) decreases.

#### Explanation:

1. **The graph crosses the x-axis:**
   - This statement is false. The exponential function \( 3^x \) never touches or crosses the x-axis. Instead, it approaches the x-axis asymptotically as \( x \) goes to negative infinity.

2. **The graph has a horizontal asymptote at \( y = 0 \):**
   - This statement is true. As \( x \) approaches negative infinity, \( 3^x \) gets closer and closer to 0 but never actually becomes 0. Hence, \( y = 0 \) is a horizontal asymptote.

3. **The graph crosses the y-axis:**
   - This statement is false. The exponential function \( 3^x \) intersects the y-axis at \( (0,1) \). It should be noted that crossing the y-axis at \( x = 0 \) means the point of intersection is not considered crossing in this context.

4. **The graph has a vertical asymptote at \( x = 0 \):**
   - This statement is false. The function \( f(x) = 3^x \) does not have a vertical asymptote. Vertical asymptotes occur in rational functions (where the denominator can be zero), but exponential functions like \( 3^x \) do not have them.

5. **As \( x \) increases, \( y \) also increases:**
   - This statement is true. The function \( 3^x \) grows exponentially which means that as \( x \) increases, \( y \) (or \( 3^x \)) increases very rapidly.

6. **As \( x \) increases,
Transcribed Image Text:### Understanding the Exponential Function \( f(x) = 3^x \) Consider the graph of the exponential function \( f(x) = 3^x \). Which of the following statements are correct? (Select all that apply.) - [ ] The graph crosses the x-axis. - [x] The graph has a horizontal asymptote at \( y = 0 \). - [ ] The graph crosses the y-axis. - [ ] The graph has a vertical asymptote at \( x = 0 \). - [x] As \( x \) increases, \( y \) also increases. - [ ] As \( x \) increases, \( y \) decreases. #### Explanation: 1. **The graph crosses the x-axis:** - This statement is false. The exponential function \( 3^x \) never touches or crosses the x-axis. Instead, it approaches the x-axis asymptotically as \( x \) goes to negative infinity. 2. **The graph has a horizontal asymptote at \( y = 0 \):** - This statement is true. As \( x \) approaches negative infinity, \( 3^x \) gets closer and closer to 0 but never actually becomes 0. Hence, \( y = 0 \) is a horizontal asymptote. 3. **The graph crosses the y-axis:** - This statement is false. The exponential function \( 3^x \) intersects the y-axis at \( (0,1) \). It should be noted that crossing the y-axis at \( x = 0 \) means the point of intersection is not considered crossing in this context. 4. **The graph has a vertical asymptote at \( x = 0 \):** - This statement is false. The function \( f(x) = 3^x \) does not have a vertical asymptote. Vertical asymptotes occur in rational functions (where the denominator can be zero), but exponential functions like \( 3^x \) do not have them. 5. **As \( x \) increases, \( y \) also increases:** - This statement is true. The function \( 3^x \) grows exponentially which means that as \( x \) increases, \( y \) (or \( 3^x \)) increases very rapidly. 6. **As \( x \) increases,
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