BackCollege Algebra Study Guide: Functions, Exponential & Logarithmic Equations, and Systems
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Functions and Their Properties
Definition of a Function
A function is a correspondence between two sets where each input from the domain is assigned exactly one output in the codomain. If is a function, denotes the output for input .
Domain: The set of all possible input values ().
Range: The set of all possible output values ().
One-to-One Function: A function is one-to-one if implies .
Example: is not one-to-one since .
Finding the Inverse of a Function
The inverse function reverses the effect of . To find the inverse:
Step 1: Replace with .
Step 2: Interchange and .
Step 3: Solve for in terms of .
Step 4: Replace with .
Example: For , the inverse is .
Composition of Functions
The composition of functions and is written as .
To find , substitute into .
Example: If and , then .
Exponential Functions
Definition and Properties
An exponential function has the form , where and .
Domain: All real numbers
Range: All positive real numbers
Graph passes through
Growth or decay depends on
Example: is an exponential growth function.
Graphing Exponential Functions
To graph :
Make a table of values for and .
Plot the points and sketch the curve.
x | |
|---|---|
-2 | 0.25 |
-1 | 0.5 |
0 | 1 |
1 | 2 |
2 | 4 |
Example: The graph of increases rapidly for positive .
Logarithmic Functions
Definition and Properties
A logarithmic function is the inverse of an exponential function. It is defined as if and only if , where , .
Domain:
Range: All real numbers
Graph passes through
Example: is the inverse of .
Properties of Logarithms
Product Rule:
Quotient Rule:
Power Rule:
Change of Base Formula:
Solving Logarithmic and Exponential Equations
To solve equations involving logarithms and exponents:
Use properties of logarithms to combine or expand expressions.
Convert between exponential and logarithmic forms.
Apply algebraic techniques to isolate the variable.
Example: Solve :
Population Growth and Exponential Models
Exponential Growth Formula
Population growth can be modeled by the formula:
= initial population
= growth rate
= time
Example: If and , then after years:
Systems of Equations
Solving Systems by Substitution and Elimination
A system of equations consists of two or more equations with the same variables. Solutions are values that satisfy all equations simultaneously.
Substitution Method: Solve one equation for one variable and substitute into the other.
Elimination Method: Add or subtract equations to eliminate a variable.
Example: Solve the system:
Add equations: Substitute: Solution:
Summary Table: Logarithm Properties
Property | Formula |
|---|---|
Product Rule | |
Quotient Rule | |
Power Rule | |
Change of Base |
Additional info:
Some examples and explanations have been expanded for clarity and completeness.
Tables have been recreated to summarize key points and provide sample values for graphing.
