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

Verified step by step guidance

1. Identify the function given: \( f(x) = 3^x \). We need to find \( f(-2) \), which means substituting \( x = -2 \) into the function.
2. Substitute \( -2 \) for \( x \) in the function: \( f(-2) = 3^{-2} \).
3. Recall the property of exponents for negative powers: \( a^{-n} = \frac{1}{a^n} \). So, \( 3^{-2} = \frac{1}{3^2} \).
4. Calculate the denominator: \( 3^2 = 3 \times 3 = 9 \), so \( f(-2) = \frac{1}{9} \).
5. If needed, convert \( \frac{1}{9} \) 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 negative and positive exponents is essential.

Evaluating Functions at Specific Inputs

Evaluating a function at a given input means substituting the input value into the function's formula and simplifying. For example, to find f(-2), replace x with -2 and calculate the result, paying attention to exponent rules, especially with negative exponents.

Rounding Decimal Values

After calculating function values, answers often need to be rounded to a specified decimal place. Rounding to the nearest thousandth means keeping three digits after the decimal point and adjusting the last digit based on the following digit to ensure accuracy and clarity.