Question
Describe the end behavior, and intervals of increase/decrease for the exponential function graphed below. (5 points) Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse. End Behavior: As x approaches −∞−∞, f(x)f(x) approaches Response area As x approaches ∞∞, f(x)f(x) approaches Response area Intervals of Increase/Decrease: The exponential function is Response area from Response area to Response area. t decreasing−∞−∞5increasing∞∞01
Graph starts at:
(0,1)
Graph starts at:
(0,1)
Answers
Answered by
GPT-4o mini
Let's analyze the end behavior and intervals of increase/decrease for the exponential function you've described, which appears to be of the form \( f(x) = a \cdot b^x \) for some \( a > 0 \) and \( b > 1 \).
### End Behavior:
1. **As \( x \) approaches \( -\infty \)**, \( f(x) \) approaches **0**. This is because exponential functions with a positive base will approach 0 as \( x \) goes to negative infinity.
2. **As \( x \) approaches \( \infty \)**, \( f(x) \) approaches **∞**. Exponential functions will grow without bound as \( x \) goes to positive infinity.
### Intervals of Increase/Decrease:
- The exponential function is **increasing** from **−∞** to **∞**. Exponential functions are always increasing when the base \( b > 1 \).
### Responses:
- End Behavior:
- As \( x \) approaches \( -\infty \), \( f(x) \) approaches **0**
- As \( x \) approaches \( \infty \), \( f(x) \) approaches **∞**
- Intervals of Increase/Decrease:
- The exponential function is **increasing** from **−∞** to **∞**.
You can input the following responses in the designated areas:
1. Response area for "As x approaches −∞, f(x) approaches": **0**
2. Response area for "As x approaches ∞, f(x) approaches": **∞**
3. Response area for "The exponential function is": **increasing**
4. Response area for "from": **−∞**
5. Response area for "to": **∞**
### End Behavior:
1. **As \( x \) approaches \( -\infty \)**, \( f(x) \) approaches **0**. This is because exponential functions with a positive base will approach 0 as \( x \) goes to negative infinity.
2. **As \( x \) approaches \( \infty \)**, \( f(x) \) approaches **∞**. Exponential functions will grow without bound as \( x \) goes to positive infinity.
### Intervals of Increase/Decrease:
- The exponential function is **increasing** from **−∞** to **∞**. Exponential functions are always increasing when the base \( b > 1 \).
### Responses:
- End Behavior:
- As \( x \) approaches \( -\infty \), \( f(x) \) approaches **0**
- As \( x \) approaches \( \infty \), \( f(x) \) approaches **∞**
- Intervals of Increase/Decrease:
- The exponential function is **increasing** from **−∞** to **∞**.
You can input the following responses in the designated areas:
1. Response area for "As x approaches −∞, f(x) approaches": **0**
2. Response area for "As x approaches ∞, f(x) approaches": **∞**
3. Response area for "The exponential function is": **increasing**
4. Response area for "from": **−∞**
5. Response area for "to": **∞**