Question

Which pair of functions consistently exhibits exponential growth at a faster rate than the quadratic function over the interval 0 < x < 5?

A) f(x) = 3x² and g(x) = 2ˣ

B) f(x) = x² and g(x) = eˣ

C) f(x) = 4x² and g(x) = 3ˣ

D) f(x) = 2x² and g(x) = 5ˣ

Answer 1

Pair B (f(x) = x² and g(x) = eˣ) consistently exhibits exponential growth at a faster rate than the quadratic function over the interval 0 < x < 5.

Step-by-step explanation:

To determine which pair of functions consistently exhibits exponential growth at a faster rate than the quadratic function over the interval 0 < x < 5, we need to compare the growth rates of the given functions.

The quadratic function has the form f(x) = ax^2, where a is a constant. Let's analyze each pair of functions:

1. Pair A: f(x) = 3x² and g(x) = 2ˣ
   Exponential function g(x) = 2ˣ grows at a faster rate than the quadratic function f(x) = 3x², because the base 2 is greater than the coefficient 3.
2. Pair B: f(x) = x² and g(x) = eˣ
   The exponential function g(x) = eˣ grows at a much faster rate than the quadratic function f(x) = x². The exponential function uses the base e, which is approximately 2.718, making its growth rate far greater than x².
3. Pair C: f(x) = 4x² and g(x) = 3ˣ
   Exponential function g(x) = 3ˣ grows at a faster rate than the quadratic function f(x) = 4x², since the base 3 is greater than the coefficient 4.
4. Pair D: f(x) = 2x² and g(x) = 5ˣ
   The exponential function g(x) = 5ˣ grows at a much faster rate than the quadratic function f(x) = 2x². The base 5 is greater than the coefficient 2, resulting in a higher growth rate.

Therefore, Pair B (f(x) = x² and g(x) = eˣ) consistently exhibits exponential growth at a faster rate than the quadratic function over the interval 0 < x < 5.