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Ch. 4 - Inverse, Exponential, and Logarithmic Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 4 - Inverse, Exponential, and Logarithmic FunctionsProblem 26
Chapter 5, Problem 26

For ƒ(x) = 3x and g(x)= (1/4)x find each of the following. Round answers to the nearest thousandth as needed. See Example 1. g(2.34)

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1
Identify the function g(x) given as \(g(x) = \left( \frac{1}{4} \right)^x\).
Substitute the given value \(x = 2.34\) into the function: \(g(2.34) = \left( \frac{1}{4} \right)^{2.34}\).
Rewrite the expression if needed, recognizing that \(\frac{1}{4} = 4^{-1}\), so \(g(2.34) = 4^{-2.34}\).
Use a calculator to evaluate \(4^{-2.34}\), which involves calculating the power of a number with a decimal exponent.
Round the result to the nearest thousandth as instructed.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 depending on whether the base is greater than or less than 1. Understanding how to evaluate these functions for any real number x is essential.
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Evaluating Functions at a Given Input

To find the value of a function at a specific input, substitute the input value into the function's formula and simplify. This process often involves calculating powers or roots, especially for exponential functions, and may require a calculator for non-integer exponents.
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Evaluating Composed Functions

Rounding to a Specified Decimal Place

Rounding is the process of limiting 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, which requires understanding place value and rounding rules.
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