5.7 Functions
3 min read•june 18, 2024
Functions are the building blocks of mathematics, describing relationships between inputs and outputs. They're everywhere in our daily lives, from calculating tips to predicting weather patterns. Understanding how to work with functions is key to solving real-world problems.
In this section, we'll explore , evaluation, and the . We'll also dive into and , which define the possible inputs and outputs of a . These concepts form the foundation for more advanced mathematical thinking.
Function Fundamentals
Function notation evaluation
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- Function notation denotes the output value () of function for input value ()
- If , then substitutes input value and simplifies
- Functions can be represented using equations, tables, or graphs
- Evaluate functions using tables by finding the output value corresponding to the given input value
- Evaluate functions using graphs by locating the point with the given input value and determining the corresponding output value
Vertical line test application
- Vertical line test determines if a graph represents a function
- Graph is a function if and only if every vertical line intersects the graph at most once
- Graph fails the vertical line test and is not a function if any vertical line intersects more than once
- Imagine drawing vertical lines at various x-values on the graph to apply the test
- A ([y = x^2](https://www.fiveableKeyTerm:y_=_x^2)) passes the vertical line test and represents a function
- A circle ([x^2 + y^2 = 1](https://www.fiveableKeyTerm:x^2_+_y^2_=_1)) fails the vertical line test and does not represent a function
Domain and Range
Domain and range identification
- is the set of all possible input values (x-values) for a function
- For graphs, domain consists of all x-values where the function is defined (has a corresponding y-value)
- For or , domain is the set of all first coordinates (x-values)
- Range is the set of all possible output values (y-values) for a function
- For graphs, range consists of all y-values that correspond to at least one x-value in the domain
- For ordered pairs or mappings, range is the set of all second coordinates (y-values) or elements in the "to" set
- For , the domain is all real numbers () and the range is all non-negative real numbers ([))
- For the function {(1, 2), (3, 4), (5, 6)}, the domain is {1, 3, 5} and the range is {2, 4, 6}
Advanced Function Concepts
Function properties and operations
- : Each element of the codomain is paired with at most one element of the domain
- : A function that "undoes" the original function, swapping input and output
- : A function with no breaks, gaps, or jumps in its graph
- : Applying one function to the result of another function (e.g., )
- : A function defined by multiple sub-functions, each applying to a specific interval of the domain
Key Terms to Review (24)
(-∞, ∞): The notation (-∞, ∞) represents the entire set of real numbers in interval notation, indicating that the values extend indefinitely in both the negative and positive directions. This concept is essential in understanding the domain and range of functions, as it shows that a function can take on any real value without restrictions. It is particularly relevant when analyzing functions that are defined for all real numbers.
[0, ∞): [0, ∞) represents the set of all real numbers that are greater than or equal to 0. This interval includes 0 and extends infinitely to the right along the number line, making it an important concept in mathematics, especially when discussing non-negative values in various functions.
Continuous function: A continuous function is a type of mathematical function where small changes in the input result in small changes in the output, meaning that the function does not have any breaks, jumps, or holes in its graph. This property makes continuous functions important in calculus and real analysis, as they can be analyzed using limits and derivatives, providing a smoother and more predictable behavior than discontinuous functions.
Dependent variable: A dependent variable is a measurable factor that responds to changes in another variable, often referred to as the independent variable. It represents the outcome or effect that researchers are interested in observing, making it crucial for understanding relationships in various mathematical contexts. Its value depends on the input from the independent variable, highlighting its role in functions, systems of equations, and statistical analysis.
Domain: The domain of a function is the set of all possible input values (typically represented as 'x') that the function can accept. It determines the range of values for which the function is defined and produces real numbers.
Domain: In mathematics, the domain refers to the complete set of possible values that a variable can take. It plays a crucial role in defining functions, inequalities, and equations as it establishes the input values for which these mathematical expressions are valid. Understanding the domain helps in determining where a function or equation can be applied and helps avoid undefined situations.
Evaluating the function: Evaluating the function means finding the output value of a function for a given input value. It involves substituting the input value into the function's equation and simplifying.
F(x): In mathematics, f(x) denotes a function where 'f' is the name of the function and 'x' represents the input value or independent variable. This notation is used to express the relationship between a set of inputs and their corresponding outputs, making it easier to understand how one quantity depends on another. The expression f(x) captures the idea that each input x can be transformed into an output based on specific rules defined by the function.
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. In mathematical notation, it is often represented as f(x).
Function composition: Function composition is the process of combining two functions to produce a new function, where the output of one function becomes the input of another. This concept allows for the creation of more complex functions by linking simpler functions together, enhancing their applicability and usefulness in various mathematical contexts. Understanding function composition is essential for analyzing relationships between different mathematical entities and for solving problems involving multiple functions.
Function notation: Function notation is a way to represent functions mathematically, using symbols to express the relationship between inputs and outputs. It typically uses the letter 'f' or other letters to denote a function, followed by parentheses containing the input variable, such as 'f(x)', which indicates that 'f' is a function of 'x'. This notation not only simplifies the representation of functions but also makes it easier to evaluate and manipulate them in various mathematical contexts.
Independent variable: An independent variable is a quantity that is manipulated or changed in an experiment or mathematical function to observe its effect on another quantity, called the dependent variable. In various contexts, it serves as the input for functions, the variable that is controlled in experiments, or the predictor in statistical models.
Inverse function: An inverse function is a function that essentially reverses the action of the original function. If you have a function 'f' that takes an input 'x' and produces an output 'y', the inverse function, denoted as 'f^{-1}', takes 'y' back to 'x'. This relationship highlights how two functions can be interconnected, where applying one after the other returns you to your starting point.
Mapping: Mapping is the process of associating each element from one set (domain) to an element in another set (codomain). It describes how elements of the domain correspond to elements of the codomain, often through a function.
Mappings: Mappings are mathematical structures that define a relationship between two sets, where each element in the first set (the domain) is paired with exactly one element in the second set (the codomain). This concept is fundamental to understanding functions, as every function can be viewed as a specific type of mapping, ensuring that each input corresponds to a unique output. Mappings not only help describe how different elements relate to each other but also play a crucial role in determining properties like continuity, injectivity, and surjectivity.
One-to-one function: A one-to-one function, or injective function, is a type of function where every element of the range is mapped from a unique element of the domain. This means that no two different inputs produce the same output, ensuring that each output value corresponds to only one input value. One-to-one functions are important because they have unique inverses, which means you can reverse the function's operation and retrieve the original input from the output.
Ordered Pairs: An ordered pair is a pair of numbers used to represent a point in a coordinate system, typically written in the form (x, y). The first number, x, indicates the position along the horizontal axis, while the second number, y, indicates the position along the vertical axis. This concept is crucial for understanding how to plot points on a graph and analyze relationships between variables in equations and functions.
Parabola: A parabola is a symmetric, U-shaped curve that is defined as the set of all points in a plane equidistant from a fixed point called the focus and a fixed line called the directrix. Parabolas can be represented mathematically by quadratic equations in two variables, typically in the form of $$y = ax^2 + bx + c$$ or $$x = ay^2 + by + c$$. This unique shape is important in various applications, such as projectile motion and reflective properties in physics.
Piecewise function: A piecewise function is a mathematical function that is defined by different expressions or formulas depending on the input value. This type of function allows for flexibility in modeling complex behaviors, as it can represent different relationships over different intervals. Piecewise functions are particularly useful in cases where a single formula cannot adequately describe the entire range of inputs.
Range: Range refers to the set of all possible output values (or dependent variable values) of a function, determined by the inputs in the domain. Understanding range is crucial as it helps to identify the limits of a function's output and how it behaves under different conditions, which can be connected to various mathematical concepts including inequalities, quadratic equations, and statistical measures.
Relation: A relation is a set of ordered pairs, where each pair consists of an input value and an output value. Relations can represent any association between elements of two sets.
Vertical line test: The vertical line test is a method used to determine if a graph represents a function. Specifically, if any vertical line drawn through the graph intersects it at more than one point, the graph does not represent a function. This test helps identify whether each input in a relation corresponds to exactly one output, ensuring that the definition of a function is met.
X^2 + y^2 = 1: The equation $$x^2 + y^2 = 1$$ represents a circle with a radius of 1 centered at the origin (0,0) in the Cartesian coordinate system. This fundamental representation connects to various aspects of geometry and functions, as it illustrates the relationship between x and y coordinates that satisfy this circular boundary. It is an essential concept in understanding circular functions and their applications in trigonometry, as well as in analyzing more complex functions that involve circles.
Y = x^2: The equation y = x^2 represents a quadratic function, which is a specific type of polynomial function where the highest exponent of the variable x is 2. This function creates a parabolic graph that opens upwards, with its vertex at the origin (0,0). The shape of the graph and its properties are essential in understanding functions, their transformations, and their applications in various mathematical contexts.