Question

Over which interval are the exponential and linear function approximately the same?

mc003-1.jpg
from 0.25 to 0.5
from 0.5 to 0.75
from 0.75 to 1.0
from 1.25 to 1.5

Answer 1

To determine the interval over which the exponential function \( y = e^x \) and the linear function \( y = x + 1 \) are approximately the same, we can analyze their behavior graphically or numerically. However, in the absence of a specific graph provided in the question, we can identify the intervals numerically or by some estimations.

We can evaluate both functions at the specified intervals:

1. **From 0.25 to 0.5:**
- \( e^{0.25} \approx 1.284 \) and \( 0.25 + 1 = 1.25 \)
- \( e^{0.5} \approx 1.648 \) and \( 0.5 + 1 = 1.5 \)

2. **From 0.5 to 0.75:**
- \( e^{0.5} \approx 1.648 \) and \( 0.5 + 1 = 1.5 \)
- \( e^{0.75} \approx 2.117 \) and \( 0.75 + 1 = 1.75 \)

3. **From 0.75 to 1.0:**
- \( e^{0.75} \approx 2.117 \) and \( 0.75 + 1 = 1.75 \)
- \( e^{1.0} \approx 2.718 \) and \( 1.0 + 1 = 2.0 \)

4. **From 1.25 to 1.5:**
- \( e^{1.25} \approx 3.490 \) and \( 1.25 + 1 = 2.25 \)
- \( e^{1.5} \approx 4.482 \) and \( 1.5 + 1 = 2.5 \)

By comparing these values, we can see that:
- Around \( x = 0.5 \), both functions are relatively close, but the exponential function starts to grow faster thereafter.
- By \( x = 0.75 \), there is a larger gap between both values and it increases further.

Based on this evaluation, the interval where the exponential and linear functions are approximately the same is most likely **from 0.25 to 0.5**.