Question

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

a) 0.25 to 0.5
b) 0.5 to 0.75
c) 0.75 to 1.0
d) None of the above

Answer 1

The exponential and linear functions are approximately the same over the interval 0.75 to 1.0.

Step-by-step explanation:

When exponential and linear functions are approximately the same, it means that their y-values are close to each other over a certain interval. To determine this interval, we can compare the values of the functions at different x-values within the given options.

Let's evaluate the functions at each interval:

a) 0.25 to 0.5: If the exponential function is y =e^x and the linear function is y = x, then e^0.25 ≈ 1.28 and 0.25 ≈ 0.25. These values are not close enough, so this interval is not the answer.

b) 0.5 to 0.75: e^0.5 ≈ 1.65 and 0.5 ≈ 0.5. These values are closer than the previous interval, but still not approximately the same.

c) 0.75 to 1.0: e^0.75 ≈ 2.12 and 0.75 ≈ 0.75. These values are very close to each other, making this interval the answer.

d) None of the above: This option is incorrect because we found that the exponential and linear functions are approximately the same over the interval 0.75 to 1.0.