82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.


2ˣ and 4ˣ⸍²

Verified step by step guidance

1. Step 1: Understand the problem by identifying the two functions given: \( f(x) = 2^x \) and \( g(x) = 4^{x^{1/2}} \). We need to determine which function grows faster as \( x \to \infty \).
2. Step 2: Simplify the second function \( g(x) = 4^{x^{1/2}} \). Notice that \( 4^{x^{1/2}} = (2^2)^{x^{1/2}} = 2^{2x^{1/2}} \). This means \( g(x) = 2^{2\sqrt{x}} \).
3. Step 3: Compare the exponents of the base 2 in both functions. For \( f(x) = 2^x \), the exponent is \( x \). For \( g(x) = 2^{2\sqrt{x}} \), the exponent is \( 2\sqrt{x} \).
4. Step 4: Analyze the growth of the exponents as \( x \to \infty \). The exponent \( x \) in \( f(x) \) grows linearly, while the exponent \( 2\sqrt{x} \) in \( g(x) \) grows as the square root of \( x \), which is slower than linear growth.
5. Step 5: Conclude that \( f(x) = 2^x \) grows faster than \( g(x) = 4^{x^{1/2}} \) as \( x \to \infty \), because the linear growth of \( x \) in the exponent of \( f(x) \) outpaces the square root growth of \( 2\sqrt{x} \) in the exponent of \( g(x) \).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions of the form f(x) = a * b^x, where 'a' is a constant, 'b' is the base, and 'x' is the exponent. These functions grow rapidly as 'x' increases, and their growth rate is determined by the base 'b'. In this question, the functions 2^x and 4^x are both exponential, but they have different bases, which affects their growth rates.

Growth Rate Comparison

To compare the growth rates of two functions, we often analyze their limits or use derivatives. For exponential functions, a common method is to express them in terms of the same base. In this case, 4^x can be rewritten as (2^2)^x = 2^(2x), allowing for a direct comparison with 2^x. This helps in determining which function grows faster as 'x' approaches infinity.

Limit Analysis

Limit analysis involves evaluating the behavior of functions as they approach a certain point, often infinity. By calculating the limit of the ratio of two functions, we can determine their relative growth rates. If the limit approaches zero, one function grows slower; if it approaches infinity, the other grows faster; and if it approaches a non-zero constant, they grow at comparable rates.