For ƒ(x) = 3^x and g(x)= (1/4)^x find each of the following. Round answers to the nearest thousandth as needed. See Example 1. g(2)

Verified step by step guidance

1. Identify the function g(x) = \left(\frac{1}{4}\right)^x.
2. Substitute x = 2 into the function g(x).
3. Calculate g(2) = \left(\frac{1}{4}\right)^2.
4. Simplify the expression \left(\frac{1}{4}\right)^2 to find the value.
5. Round the result to the nearest thousandth as needed.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in the form f(x) = a^x, where 'a' is a positive constant. These functions exhibit rapid growth or decay depending on the base 'a'. In this question, f(x) = 3^x and g(x) = (1/4)^x are both exponential functions, with g(x) representing exponential decay since its base is a fraction less than 1.

Evaluating Functions

Evaluating a function involves substituting a specific value for the variable in the function's expression. For example, to find g(2), you replace 'x' in g(x) = (1/4)^x with 2, resulting in g(2) = (1/4)^2. This process is essential for determining the output of the function at a given input.

Rounding Numbers

Rounding numbers is the process of adjusting a number to a specified degree of accuracy, often to make it simpler or more understandable. In this context, rounding to the nearest thousandth means adjusting the result of g(2) to three decimal places. This is important for presenting answers in a clear and standardized format.