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. Recall the property of exponents: \(a^{-n} = \frac{1}{a^n}\). Apply this to rewrite \(\left( \frac{1}{4} \right)^{-3}\) as \$4^3$.
4. Calculate \$4^3\( by multiplying 4 by itself three times: \)4 \times 4 \times 4$.
5. After finding the value, round the result to the nearest thousandth if necessary.

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 given x is essential.

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the positive exponent, i.e., a^(-x) = 1/(a^x). This concept is crucial when evaluating functions like g(x) = (1/4)^x at negative values of x.

Rounding to a Specified Decimal Place

Rounding involves approximating a number to a certain number of decimal places, here to the nearest thousandth (three decimal places). This ensures answers are presented clearly and consistently, especially when dealing with irrational or long decimal values.