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

Verified step by step guidance

1. Identify the function g(x) given as \(g(x) = \left( \frac{1}{4} \right)^x\).
2. Substitute the value \(x = 2\) into the function to find \(g(2)\), so write \(g(2) = \left( \frac{1}{4} \right)^2\).
3. Recall that raising a fraction to a power means raising both numerator and denominator to that power: \(\left( \frac{1}{4} \right)^2 = \frac{1^2}{4^2}\).
4. Simplify the expression to get \(\frac{1}{16}\).
5. If needed, convert the fraction to a decimal and round to the nearest thousandth.

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 g(2).

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(2), replace x with 2 in g(x) and calculate the result. This process is fundamental for interpreting and solving function-related questions.

Rounding Decimal Numbers

Rounding involves approximating a number to a specified decimal place to simplify results. In this problem, answers must be rounded to the nearest thousandth, meaning three digits after the decimal point. Proper rounding ensures clarity and consistency in presenting numerical answers.