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(3)

Verified step by step guidance

1. Identify the function g(x) given as \(g(x) = \left( \frac{1}{4} \right)^x\).
2. To find \(g(3)\), substitute \(x = 3\) into the function: \(g(3) = \left( \frac{1}{4} \right)^3\).
3. Rewrite the expression using exponent rules: \(\left( \frac{1}{4} \right)^3 = \frac{1^3}{4^3} = \frac{1}{4^3}\).
4. Calculate the denominator \$4^3\( by multiplying 4 by itself three times: \)4 \times 4 \times 4$.
5. Express the final value as a fraction or decimal, then round to the nearest thousandth as required.

Key Concepts

Exponential Functions

Exponential functions have the form f(x) = a^x, where the base a is a positive constant. They model growth or decay processes and are evaluated by raising the base to the power of the input x. Understanding how to compute values for these functions is essential for solving problems like finding g(3).

Evaluating Functions at a Given Input

Evaluating a function at a specific input means substituting the input value into the function's formula and simplifying. For example, to find g(3), replace x with 3 in g(x) and calculate the result. This process is fundamental for interpreting and solving function-related questions.

Rounding to a Specified Decimal Place

Rounding involves adjusting a number to a certain number of decimal places for simplicity or clarity. In this problem, answers must be rounded to the nearest thousandth, meaning three digits after the decimal point. Proper rounding ensures answers are precise and consistent with instructions.