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(-5/2)

Verified step by step guidance

1. Identify the function g(x) given as \(g(x) = \left( \frac{1}{4} \right)^x\).
2. Substitute the given value \(x = -\frac{5}{2}\) into the function: \(g\left(-\frac{5}{2}\right) = \left( \frac{1}{4} \right)^{-\frac{5}{2}}\).
3. Recall the property of exponents: \(a^{-b} = \frac{1}{a^b}\). Use this to rewrite the expression as \(\left( \frac{1}{4} \right)^{-\frac{5}{2}} = 4^{\frac{5}{2}}\).
4. Express \$4^{\frac{5}{2}}\( as \)\left(4^{\frac{1}{2}}\right)^5\( or \)\left(\sqrt{4}\right)^5$ to simplify the calculation.
5. Calculate \(\sqrt{4}\), then raise the result to the 5th power, and round the final answer 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 have unique properties such as a constant ratio between successive outputs. Understanding how to evaluate these functions for any real exponent is essential.

Negative and Fractional Exponents

Negative exponents represent the reciprocal of the base raised to the positive exponent, e.g., a^(-n) = 1/a^n. Fractional exponents correspond to roots, such as a^(1/2) = √a. Evaluating functions at negative or fractional inputs requires applying these rules correctly.

Rounding and Approximation

When calculating values that result in irrational or long decimal numbers, rounding to a specified decimal place is necessary. Rounding to the nearest thousandth means keeping three digits after the decimal point, ensuring answers are both precise and manageable.