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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2:40 minutes

Problem 1

Textbook Question

Fill in the blank(s) to correctly complete each sentence. If ƒ(x) = 4^x, then ƒ(2) = and ƒ(-2) = ________.

Verified step by step guidance

Identify the function given: $ f(x) = 4^x $.

To find $ f(2) $, substitute $ x = 2 $ into the function: $ f(2) = 4^2 $.

Calculate $ 4^2 $ to find $ f(2) $.

To find $ f(-2) $, substitute $ x = -2 $ into the function: $ f(-2) = 4^{-2} $.

Calculate $ 4^{-2} $ by finding the reciprocal of $ 4^2 $.

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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 ƒ(x) = a^x, where 'a' is a positive constant and 'x' is the variable. These functions exhibit rapid growth or decay depending on the base 'a'. In this case, ƒ(x) = 4^x represents an exponential function where the base is 4, indicating that as 'x' increases, the value of ƒ(x) increases exponentially.

Evaluating Functions

Evaluating a function involves substituting a specific value for the variable in the function's expression. For example, to find ƒ(2) in the function ƒ(x) = 4^x, you replace 'x' with 2, resulting in ƒ(2) = 4^2 = 16. Similarly, for ƒ(-2), you substitute -2 for 'x', leading to ƒ(-2) = 4^(-2) = 1/16, demonstrating how to compute function values for both positive and negative inputs.

Properties of Exponents

Properties of exponents are rules that govern how to manipulate exponential expressions. Key properties include the product of powers (a^m * a^n = a^(m+n)), the power of a power ( (a^m)^n = a^(m*n)), and the negative exponent rule (a^(-n) = 1/a^n). Understanding these properties is essential for simplifying expressions and solving equations involving exponents, such as those found in the given function.