Functions are all about relationships between inputs and outputs. In this topic, we dive into domains and ranges, which set the rules for what values can go in and come out of a . Understanding these concepts helps us grasp how functions behave and what they can represent.
Real-world applications bring these ideas to life. We'll explore how domains and ranges apply to things like population growth, pricing, and motion. This practical angle shows how math concepts directly connect to everyday situations and problem-solving.
Domain and Range of Functions
Restrictions on input values
Definition of : set of all possible input values (usually x) for which a function is defined
Identifying restrictions on the domain
Denominator cannot equal zero in rational functions, find values of x that make the denominator zero and exclude them from the domain
Square root of a negative number is undefined, ensure the radicand (expression under the square root) is non-negative for functions with square roots
Logarithms are only defined for positive arguments, the input must be greater than zero for logarithmic functions
: {x∣condition} defines the set of all elements x that satisfy a given condition
Domain from function graphs
Identifying domain from graphs: consists of all x-values for which the graph is defined
Vertical lines represent undefined values and are not part of the domain (asymptotes)
Identifying range from graphs: consists of all y-values that the function takes on
Horizontal lines represent y-values that are not part of the range (asymptotes)
Types of functions and their typical domains and ranges
Linear functions: are typically all real numbers (R)
Quadratic functions: domain is all real numbers, range depends on the direction of the parabola (upward or downward)
Exponential functions: domain is all real numbers, range is typically positive real numbers (R+)
Logarithmic functions: domain is positive real numbers, range is all real numbers
Continuous functions have a domain and range that form unbroken intervals
Real-world applications of domain
Identifying the domain and range in context
Determine what the input and output variables represent in the given context (time, distance, population)
Consider any real-world limitations on the input and output values (non-negative, integer values)
For discrete functions, the domain and range consist of specific, separate values
Solving problems using domain and range
Use the domain to determine the allowable input values for the function in the given context
Use the range to interpret the possible output values and their meaning in the context
Examples of real-world applications
Population growth models: domain is time (non-negative), range is population size (non-negative integers)
Supply and demand curves: domain is quantity (non-negative), range is price (non-negative)
Projectile motion: domain is time (non-negative), range is height (can be negative or positive)
Functions and Relations
A function is a special type of relation where each input value corresponds to exactly one output value
The codomain is the set of all possible output values for a function
A relation is a set of ordered pairs that describes a relationship between two sets
Mapping refers to the process of assigning output values to input values in a function
Key Terms to Review (11)
Absolute value: Absolute value of a number is its distance from zero on the number line, regardless of direction. It is always a non-negative number.
Absolute value function: An absolute value function is a piecewise function that describes the distance of a number from zero on the number line, without considering direction. It is typically written as $f(x) = |x|$, where $|x|$ denotes the absolute value of $x$.
Constant function: A constant function is a function that always returns the same value regardless of the input. It can be represented as $f(x) = c$, where $c$ is a constant.
Domain: The domain of a function is the set of all possible input values (typically $x$-values) for which the function is defined. It represents the range of values over which the function can be evaluated.
Domain and range: The domain of a function is the set of all possible input values (typically x-values) for which the function is defined. The range of a function is the set of all possible output values (typically y-values) that the function can produce.
Function: A function is a relation between a set of inputs and a set of permissible outputs where each input is related to exactly one output. It is often represented as $f(x)$.
Magnitude: Magnitude represents the size or length of a vector. It is always a non-negative value and can be found using the Pythagorean theorem in two or three dimensions.
Modulus: The modulus of a complex number is its distance from the origin in the complex plane, represented as $|z|$. It is calculated using the formula $|z| = \sqrt{a^2 + b^2}$ where $z = a + bi$.
Ordered pair: An ordered pair is a set of two numbers written in a specific order, usually as (x, y), which represents a point in the rectangular coordinate system. The first number corresponds to the x-coordinate and the second to the y-coordinate.
Piecewise function: A piecewise function is a function composed of multiple sub-functions, each defined on a specific interval of the domain. These sub-functions can vary in form and are typically separated by different conditions.
Set-builder notation: Set-builder notation is a mathematical shorthand used to describe a set by stating the properties that its members must satisfy. It is often used to specify domains and ranges of functions.