Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

6. Exponential & Logarithmic Functions

Introduction to Exponential Functions

College Algebra

6. Exponential & Logarithmic Functions

Introduction to Exponential Functions

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1:50 minutes

Problem 25

Textbook Question

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(-1.68)

Verified step by step guidance

Identify the function g(x) = \left(\frac{1}{4}\right)^x.

Substitute x = -1.68 into the function g(x).

Rewrite the expression \left(\frac{1}{4}\right)^{-1.68} using the property of exponents: a^{-b} = \frac{1}{a^b}.

Calculate \left(\frac{1}{4}\right)^{-1.68} as 4^{1.68}.

Use a calculator to evaluate 4^{1.68} and round the result to the nearest thousandth.

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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 are mathematical expressions in the form f(x) = a^x, where 'a' is a positive constant. They exhibit rapid growth or decay depending on whether 'a' is greater than or less than one. Understanding the behavior of these functions is crucial for evaluating expressions like g(-1.68) in the given problem.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the absolute value of the exponent. For example, a^(-n) = 1/(a^n). This concept is essential for evaluating g(x) = (1/4)^x, especially when x is negative, as it transforms the expression into a fraction that can be computed more easily.

Rounding Numbers

Rounding numbers involves adjusting a numerical value to a specified degree of accuracy, often to simplify calculations or present results clearly. In this context, rounding to the nearest thousandth means keeping three decimal places, which is important for providing precise answers in mathematical problems.