Question
f(x)= 2^x+1
Using the interval (-2,2)
Using the interval (-2,2)
Answers
Answered by
GPT-4o mini
To analyze the function \( f(x) = 2^x + 1 \) on the interval \((-2, 2)\), we can evaluate various aspects of the function, including its value at critical points, any points of interest within the interval, and a general sense of its behavior.
### Step 1: Evaluate the Function at Key Points
Let's evaluate \( f(x) \) at the endpoints of the interval and at some point in between:
1. **At \( x = -2 \):**
\[
f(-2) = 2^{-2} + 1 = \frac{1}{4} + 1 = \frac{5}{4} \approx 1.25
\]
2. **At \( x = 0 \):**
\[
f(0) = 2^0 + 1 = 1 + 1 = 2
\]
3. **At \( x = 2 \):**
\[
f(2) = 2^2 + 1 = 4 + 1 = 5
\]
### Step 2: Understand the Behavior of the Function
1. **Increasing/Decreasing:**
The function \( f(x) = 2^x + 1 \) is an exponential function with a base greater than 1. Exponential functions of this form are strictly increasing for all \( x \). Therefore, \( f(x) \) increases throughout the interval \((-2, 2)\).
2. **Continuity:**
The function \( f(x) \) is continuous for all real numbers, including the interval \((-2, 2)\).
### Step 3: Summary of Values
- \( f(-2) \approx 1.25 \)
- \( f(0) = 2 \)
- \( f(2) = 5 \)
### Conclusion
On the interval \((-2, 2)\), \( f(x) = 2^x + 1 \) is a continuous and strictly increasing function. The minimum value of \( f(x) \) in this interval occurs at \( x = -2 \), where \( f(-2) = \frac{5}{4} \), and the maximum value occurs at \( x = 2 \), where \( f(2) = 5 \).
### Step 1: Evaluate the Function at Key Points
Let's evaluate \( f(x) \) at the endpoints of the interval and at some point in between:
1. **At \( x = -2 \):**
\[
f(-2) = 2^{-2} + 1 = \frac{1}{4} + 1 = \frac{5}{4} \approx 1.25
\]
2. **At \( x = 0 \):**
\[
f(0) = 2^0 + 1 = 1 + 1 = 2
\]
3. **At \( x = 2 \):**
\[
f(2) = 2^2 + 1 = 4 + 1 = 5
\]
### Step 2: Understand the Behavior of the Function
1. **Increasing/Decreasing:**
The function \( f(x) = 2^x + 1 \) is an exponential function with a base greater than 1. Exponential functions of this form are strictly increasing for all \( x \). Therefore, \( f(x) \) increases throughout the interval \((-2, 2)\).
2. **Continuity:**
The function \( f(x) \) is continuous for all real numbers, including the interval \((-2, 2)\).
### Step 3: Summary of Values
- \( f(-2) \approx 1.25 \)
- \( f(0) = 2 \)
- \( f(2) = 5 \)
### Conclusion
On the interval \((-2, 2)\), \( f(x) = 2^x + 1 \) is a continuous and strictly increasing function. The minimum value of \( f(x) \) in this interval occurs at \( x = -2 \), where \( f(-2) = \frac{5}{4} \), and the maximum value occurs at \( x = 2 \), where \( f(2) = 5 \).