3.1 Functions and Function Notation
3 min read•june 24, 2024
Functions are the backbone of algebra, assigning unique outputs to inputs. They're like vending machines: you put in a value, and it spits out exactly one result. This concept is crucial for modeling real-world relationships and solving complex problems.
In this section, we'll explore notation, evaluation, and analysis. We'll also dive into different types of functions, their graphs, and key properties. Understanding these concepts will help you tackle more advanced mathematical challenges down the road.
Functions and Function Notation
Functions vs general relations
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- assigns exactly one value to each value
- can have multiple outputs for a single input, functions cannot
- Functions are a subset of relations that satisfy the
- Each input value is paired with exactly one output value
- No two ordered pairs in a function can have the same input value with different output values
Evaluation of function inputs
- Function notation: represents output value of function when input is
- If and , then
- Evaluating functions involves substituting given input value for variable in function
- Simplify expression to find corresponding output value
- Output value represents result of applying function to given input
- In context, output value can have specific meaning or interpretation (temperature, cost)
One-to-one function analysis
- has each output value corresponding to exactly one input value
- No two different input values can produce the same output value
- Determining if function is one-to-one involves checking if it passes
- If any horizontal line intersects graph of function more than once, function is not one-to-one
- One-to-one functions have inverses that are also functions
- Essential for solving equations and modeling certain real-world situations (encryption, decryption)
Vertical line test for functions
- Graphical method to determine if is a function
- If any vertical line intersects graph more than once, relation is not a function
- Applying involves sketching a few vertical lines crossing graph at different points
- Check if any vertical lines intersect graph more than once
- If all vertical lines intersect graph at most once, graph represents a function
Graphs of common functions
- Linear functions:
- Graphs as straight lines with and
- rate of change (slope) between any two points on line
- Quadratic functions:
- Graphs as parabolas, symmetric curves with a (turning point)
- Shape of depends on value of (positive or negative)
- and vertex can be found using
- Exponential functions:
- Graphs as curves that increase or decrease rapidly based on value of
- If , function increases exponentially; if , function decreases exponentially
- y-intercept is always , graph approaches x-axis as approaches negative infinity (assuming )
- Graphs as curves that increase or decrease rapidly based on value of
- Interpreting graphs involves identifying key features (intercepts, vertices, asymptotes)
- Analyze behavior of function (, , intervals)
- Relate graphical representation to algebraic form and real-world context, if applicable (population growth, radioactive decay)
- Piecewise functions: defined by different formulas over different intervals of the
Function Properties and Operations
- : set of all possible input values for a function
- : set of all possible output values of a function
- : set of all possible output values, including those not actually produced by the function
- : combining two functions to create a new function, denoted as (f ∘ g)(x) = f(g(x))
Key Terms to Review (50)
Absolute value function: An absolute value function is a type of piecewise function that returns the non-negative value of its input. It is denoted as $f(x) = |x|$ and has a V-shaped graph.
Asymptote: An asymptote is a line or curve that a graph approaches but never touches. It represents the limit of a function's behavior as the input variable approaches a particular value. Asymptotes are an important concept in various mathematical topics, including rational expressions, functions, rational functions, exponential functions, logarithmic functions, and exponential and logarithmic models.
Axis of symmetry: The axis of symmetry is a vertical line that divides a parabola into two mirror-image halves. It passes through the vertex of the parabola and has the equation $x = -\frac{b}{2a}$ for a quadratic function in standard form.
Axis of Symmetry: The axis of symmetry is a line that divides a symmetrical figure, such as a parabola or absolute value function, into two equal halves. It represents the midpoint or line of reflection for the function, where the left and right sides are mirror images of each other.
Co-vertex: The co-vertices of an ellipse are the endpoints of the minor axis. They are perpendicular to and lie at the midpoint of the major axis.
Codomain: The codomain of a function is the set of all possible output values that the function can produce. It represents the full range of values that the function is capable of mapping its input values to.
Composition: Composition refers to the way in which elements or parts are combined or arranged to form a whole. It is a fundamental concept in mathematics, particularly in the context of functions and their relationships.
Composition of functions: The composition of functions is the application of one function to the results of another. It is denoted as $(f \circ g)(x) = f(g(x))$.
Constant: A constant is a value that does not change. In algebra, it is often a number without any variables attached to it.
Constant: A constant is a fixed value that does not change within a given context or problem. It is a fundamental quantity that remains the same regardless of the circumstances or variables involved.
Decreasing: Decreasing refers to the act or process of becoming smaller, lower, or less in amount, degree, or intensity over time. It is a fundamental concept in the study of functions and function notation, where it describes the behavior of a function as it moves from one point to another.
Decreasing linear function: A decreasing linear function is a linear function where the value of the function decreases as the input increases. It has a negative slope.
Domain: The domain of a function is the complete set of possible input values (x-values) that allow the function to work within its constraints. It specifies the range of x-values for which the function is defined.
Domain: The domain of a function refers to the set of input values for which the function is defined. It represents the range of values that the independent variable can take on, and it is the set of all possible values that can be plugged into the function to produce a meaningful output.
Equation: An equation is a mathematical statement that asserts the equality of two expressions. It consists of two expressions separated by an equals sign ($=$).
Exponential function: An exponential function is a mathematical expression in the form $f(x) = a \cdot b^x$, where $a$ is a constant, $b$ is the base greater than 0 and not equal to 1, and $x$ is the exponent. These functions model growth or decay processes.
Exponential Function: An exponential function is a mathematical function in which the independent variable appears as an exponent. These functions exhibit a characteristic curve that grows or decays at a rate proportional to the current value, leading to rapid changes in output as the input increases.
F(x): f(x) is a function notation that represents a relationship between an independent variable, x, and a dependent variable, f. It is a fundamental concept in mathematics that underpins the study of functions, their properties, and their applications across various mathematical topics.
Formula: A formula is a mathematical expression that describes the relationship between different quantities. It often includes variables, constants, and arithmetic operations.
Function: A function is a relation between a set of inputs and a set of permissible outputs such that each input is related to exactly one output. Functions are often represented by $f(x)$ where $x$ is the input variable.
Function: A function is a mathematical relationship between two or more variables, where one variable (the dependent variable) depends on the value of the other variable(s) (the independent variable(s)). Functions are a fundamental concept in mathematics and are essential in understanding various topics in college algebra, including coordinate systems, quadratic equations, polynomial functions, and modeling using variation.
General form of a quadratic function: The general form of a quadratic function is expressed as $ax^2 + bx + c = 0$ where $a$, $b$, and $c$ are constants and $a \neq 0$. This representation is crucial for solving quadratic equations and analyzing their properties.
Horizontal line test: The horizontal line test is a method used to determine if a function is one-to-one (injective). If any horizontal line intersects the graph of the function at most once, then the function passes the test and is one-to-one.
Horizontal Line Test: The horizontal line test is a method used to determine whether a function is one-to-one, meaning that each output value is associated with only one input value. It involves drawing horizontal lines across the graph of a function to see if the line intersects the graph at more than one point.
Increasing: Increasing refers to the state or process of becoming greater in size, amount, or degree over time. In the context of functions and function notation, increasing describes a relationship where the output values of a function grow larger as the input values become greater.
Input: Input is the value substituted into a function. It is commonly denoted as $x$ in functional notation.
Inverse: An inverse is a mathematical operation that reverses the effect of another operation. It is a fundamental concept in functions, where the inverse function undoes the original function, restoring the original input value.
Linear Function: A linear function is a mathematical function that represents a straight line on a coordinate plane. It is characterized by a constant rate of change, known as the slope, and can be expressed in the form $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.
One-to-one function: A one-to-one function (injective function) is a function where each element of the domain maps to a unique element in the codomain. No two different elements in the domain map to the same element in the codomain.
One-to-One Function: A one-to-one function, also known as an injective function, is a function where each element in the domain is mapped to a unique element in the codomain. This means that for every input value, there is only one corresponding output value, and no two input values can be mapped to the same output value.
Ordered pair: An ordered pair is a set of two elements written in a specific order, typically as (x, y), where x represents the horizontal coordinate and y represents the vertical coordinate.
Output: The output is the result of a function for a given input value. It represents the value that the function assigns to each element in its domain.
Parabola: A parabola is a symmetric curve that represents the graph of a quadratic function. It can open upward or downward depending on the sign of the quadratic coefficient.
Parabola: A parabola is a curved, U-shaped line that is the graph of a quadratic function. It is one of the fundamental conic sections, along with the circle, ellipse, and hyperbola, and has many important applications in mathematics, science, and engineering.
Piecewise function: A piecewise function is defined by multiple sub-functions, each applying to a specific interval of the domain. It allows for different behaviors over different parts of the domain.
Piecewise Function: A piecewise function is a function that is defined by different formulas or expressions over different intervals or domains of the independent variable. These functions are constructed by 'piecing together' multiple simpler functions to create a more complex overall function.
Point-slope formula: The point-slope formula is a method used to find the equation of a line given a point on the line and its slope. It is expressed as $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope.
Quadratic Function: A quadratic function is a polynomial function of degree two, where the highest exponent of the independent variable is two. Quadratic functions are widely used in various mathematical and scientific applications, including physics, engineering, and economics.
Range: The range of a function is the set of all possible output values it can produce. It corresponds to the y-values in the coordinate plane.
Range: In mathematics, the range refers to the set of all possible output values (dependent variable values) that a function can produce based on its input values (independent variable values). Understanding the range helps in analyzing how a function behaves and what values it can take, connecting it to various concepts like transformations, compositions, and types of functions.
Relation: A relation is a set of ordered pairs where each element from the first set is associated with an element from the second set. It describes how elements from two sets are connected.
Relation: A relation is a set of ordered pairs that represents a connection or association between elements. It describes how elements in one set, called the domain, are linked to elements in another set, called the codomain.
Slope: Slope is a measure of the steepness or incline of a line or a surface. It represents the rate of change between two variables, typically the change in the vertical direction (y-coordinate) with respect to the change in the horizontal direction (x-coordinate).
Uniqueness Property: The uniqueness property is a fundamental characteristic of functions that states each input in the domain of a function is associated with exactly one output in the codomain. This property ensures that a function maps each element in the domain to a unique corresponding element in the codomain.
Vertex: The vertex is a critical point in various mathematical functions and geometric shapes. It represents the point of maximum or minimum value, or the point where a curve changes direction. This term is particularly important in the context of quadratic equations, functions, absolute value functions, and conic sections such as the ellipse and parabola.
Vertical line test: The vertical line test is a method used to determine if a graph represents a function. If any vertical line intersects the graph at more than one point, the graph does not represent a function.
Vertical Line Test: The vertical line test is a graphical method used to determine whether a relation represents a function. It involves drawing vertical lines on the coordinate plane to check if each vertical line intersects the graph at no more than one point.
X-intercept: The x-intercept is the point where a graph crosses the x-axis, where the y-coordinate is zero. It represents the solution(s) to an equation when $y = 0$.
X-Intercept: The x-intercept of a graph is the point where the graph of a function or equation intersects the x-axis, indicating the value of x when the function's output or the equation's value is zero. The x-intercept is a crucial concept in understanding the behavior and properties of various mathematical functions and equations.
Y-intercept: The y-intercept is the point at which a line or curve intersects the y-axis, representing the value of the dependent variable (y) when the independent variable (x) is zero. It is a crucial concept in understanding the behavior and properties of various mathematical functions and their graphical representations.