1.1 Algebraic Functions and Their Graphs
3 min readโขaugust 7, 2024
are the building blocks of calculus. They help us model real-world situations and understand relationships between variables. From simple linear equations to complex polynomials, these functions form the foundation for more advanced mathematical concepts.
Graphs are visual representations of functions, showing how inputs relate to outputs. They're crucial for analyzing function behavior, identifying key features like intercepts and asymptotes, and solving problems. Mastering these basics sets you up for success in calculus.
Functions and Their Properties
Defining Functions and Their Components
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- Function maps each input value from the to exactly one output value in the
- Domain consists of all possible input values (x-values) for a function
- Range comprises all possible output values (y-values) that result from the function
- combines two or more functions to create a new function, where the output of one function becomes the input of the next (e.g., )
- , denoted as , "undoes" the original function by mapping the output values back to their corresponding input values
- For a function and its inverse ,
Properties of Functions
- Functions can be one-to-one (injective) if each element in the codomain is mapped to by at most one element in the domain
- One-to-one functions have inverses that are also functions
- Functions can be onto (surjective) if every element in the codomain is mapped to by at least one element in the domain
- Functions can be both one-to-one and onto (bijective) if every element in the codomain is mapped to by exactly one element in the domain
- Functions can be even if for all in the domain or odd if for all in the domain
- Examples of even functions: ,
- Examples of odd functions: ,
Types of Functions
Linear and Polynomial Functions
- has the form , where is the and is the
- Graphs as a straight line (e.g., )
- has the form , where , , and are constants and
- Graphs as a parabola (e.g., )
- has the form , where are constants and is a non-negative integer
- Degree of the polynomial is the highest power of (e.g., is a 3rd-degree polynomial)
Rational, Piecewise, and Absolute Value Functions
- is a ratio of two polynomial functions, , where and are polynomial functions and
- Graphs with asymptotes and holes (e.g., )
- is defined by different equations over different intervals of the domain
- Graphs with distinct segments (e.g., )
- has the form , which gives the distance of from zero on the real number line
- Graphs as a V-shaped curve (e.g., )
Key Terms to Review (29)
Absolute Value Function: The absolute value function is a mathematical function that measures the distance of a number from zero on the number line, regardless of direction. It is denoted as $$f(x) = |x|$$, where the output is always non-negative, making it a piecewise function with two cases: when $$x$$ is positive or zero, and when $$x$$ is negative. This function plays an important role in understanding inequalities, graphing, and transformations.
Algebraic Functions: Algebraic functions are mathematical expressions that involve a variable and can be formed using algebraic operations such as addition, subtraction, multiplication, division, and taking roots. These functions can be represented as polynomials, rational functions, and radical functions, and they play a crucial role in understanding relationships between variables in various mathematical contexts. The graphs of algebraic functions illustrate their behavior and characteristics, which include features like intercepts, asymptotes, and end behavior.
Asymptote: An asymptote is a line that a graph approaches but never actually touches or intersects. This concept is crucial for understanding the behavior of various functions, especially as they extend towards infinity or exhibit discontinuities. Asymptotes can be horizontal, vertical, or oblique, each revealing different characteristics of the function's graph and its limits.
Bijective Function: A bijective function is a type of function that is both injective (one-to-one) and surjective (onto), meaning every element in the domain maps to a unique element in the codomain and every element in the codomain has a pre-image in the domain. This characteristic makes bijective functions particularly important because they establish a one-to-one correspondence between two sets, which allows for the existence of an inverse function. Understanding bijective functions is crucial for analyzing algebraic functions and their graphs, as they determine the function's behavior and the relationships between input and output values.
Cartesian Coordinates: Cartesian coordinates are a system that uses two or three perpendicular axes to define the position of a point in a space. This method allows for the representation of points using ordered pairs in two dimensions or ordered triples in three dimensions, connecting geometric and algebraic concepts through graphs, vectors, and various mathematical operations.
Complex roots: Complex roots are solutions to polynomial equations that include imaginary numbers, typically expressed in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, defined as $ extit{i} = \sqrt{-1}$. These roots arise when the polynomial does not intersect the x-axis, indicating that it has no real solutions. The presence of complex roots indicates important characteristics about the polynomial, including symmetry in the roots for polynomials with real coefficients.
Continuity: Continuity refers to the property of a function that ensures it does not have any abrupt changes, jumps, or holes in its graph. A function is continuous if, intuitively, you can draw its graph without lifting your pencil from the paper. This concept is crucial in understanding limits, differentiability, and the behavior of functions across different scenarios.
Domain: In mathematics, the domain refers to the complete set of possible values of the independent variable(s) for which a function is defined. It essentially outlines the input values that can be plugged into a function without leading to any undefined or nonsensical results. Understanding the domain is crucial because it helps identify the valid inputs, ensuring that the outputs of the function make sense within its context.
Even Function: An even function is a type of function that exhibits symmetry about the y-axis. This means that for every point (x, f(x)) on the graph of the function, there exists a corresponding point (-x, f(x)). This property makes even functions particularly interesting because they behave predictably when transformed or manipulated, allowing for easier analysis and graphing.
Function Composition: Function composition is the process of combining two functions, where the output of one function becomes the input of another. This operation creates a new function that represents the combined effect of both functions, often denoted as (f โฆ g)(x) = f(g(x)). Understanding function composition is essential for analyzing complex relationships between variables and is fundamental to calculus concepts such as derivatives and integrals.
Graph transformations: Graph transformations refer to the various operations that can be performed on the graph of a function to change its position, shape, or orientation. These transformations include shifting, reflecting, stretching, and compressing the graph, which allows for a visual understanding of how alterations in the function's equation affect its graphical representation.
Intercept: An intercept is a point where a graph intersects an axis on a coordinate plane. It plays a crucial role in understanding the behavior of algebraic functions, as it helps to identify key values like where the function equals zero (x-intercept) or where it reaches a specific output (y-intercept). The concept of intercepts can also indicate trends in data and solutions to equations, making them essential for graphing and analyzing functions.
Intermediate Value Theorem: The Intermediate Value Theorem states that for any continuous function defined on a closed interval, if the function takes on two values at the endpoints, then it also takes on every value between those two values at least once. This concept is crucial for understanding how functions behave, particularly in identifying the existence of roots and analyzing continuity.
Inverse Function: An inverse function is a function that reverses the effect of the original function, mapping outputs back to their corresponding inputs. For a function $$f(x)$$, its inverse, denoted as $$f^{-1}(x)$$, satisfies the property that if $$f(a) = b$$, then $$f^{-1}(b) = a$$. This concept is crucial for understanding relationships between variables and their graphical representations, as the graphs of inverse functions are reflections of each other over the line $$y = x$$.
Linear Function: A linear function is a type of function that can be graphically represented as a straight line. It is defined by the equation $$f(x) = mx + b$$, where $$m$$ represents the slope of the line and $$b$$ denotes the y-intercept. Linear functions exhibit constant rates of change, making them essential for modeling relationships in various contexts, from physics to economics.
Odd Function: An odd function is a type of function that exhibits symmetry about the origin, meaning that for every point (x, f(x)) on the graph, there exists a corresponding point (-x, -f(x)). This property leads to unique characteristics when analyzing the function's behavior and its graphical representation. Odd functions play a crucial role in calculus, particularly in understanding integrals and series expansions.
One-to-One Function: A one-to-one function is a type of function where each input is mapped to a unique output, meaning no two different inputs produce the same output. This property ensures that the function is invertible, allowing for the establishment of a clear relationship between the input and output values. Identifying one-to-one functions is crucial in understanding the behavior of algebraic functions and their graphs, particularly when determining if a function has an inverse that is also a function.
Onto Function: An onto function, also known as a surjective function, is a type of function where every element in the codomain is mapped to by at least one element from the domain. This means that for a function to be onto, there cannot be any elements in the codomain that are left out; every possible output must have a corresponding input. Understanding onto functions is essential when analyzing the behavior of algebraic functions and their graphs, as it helps in determining whether a function can cover its entire range.
Piecewise Function: A piecewise function is a function defined by multiple sub-functions, each applying to a specific interval of the domain. These functions allow for different rules or expressions to be used depending on the input value, making them useful for modeling situations that have distinct behaviors in different ranges. This concept is crucial in various mathematical analyses, particularly when dealing with discontinuities, integration, and geometric properties.
Polynomial function: A polynomial function is a mathematical expression that involves variables raised to whole number powers, combined using addition, subtraction, and multiplication. These functions can have one or more terms, where each term consists of a coefficient and a variable raised to a non-negative integer exponent. Polynomial functions are crucial in understanding the behavior of graphs, roots, and various applications in calculus and algebra.
Quadratic function: A quadratic function is a type of polynomial function of degree two, typically expressed in the standard form $$f(x) = ax^2 + bx + c$$ where $$a$$, $$b$$, and $$c$$ are constants and $$a$$ is not equal to zero. Quadratic functions have a distinct U-shaped graph called a parabola, which can open upwards or downwards depending on the sign of the leading coefficient $$a$$. The vertex of the parabola represents the maximum or minimum point of the function and is crucial for understanding its behavior.
Range: The range of a function refers to the set of all possible output values (or dependent variables) that can result from plugging in values from the domain (or independent variables) into the function. Understanding the range helps in visualizing how a function behaves and where its values can lie, providing insight into important characteristics such as maximum and minimum points, which are crucial when analyzing algebraic and exponential/logarithmic functions.
Rational Function: A rational function is a type of function that can be expressed as the quotient of two polynomial functions, where the denominator is not equal to zero. These functions can have a variety of behaviors and features, including asymptotes, intercepts, and end behavior, which are essential in understanding their graphs and how they behave in different situations.
Real Roots: Real roots refer to the solutions of a polynomial equation that are real numbers, meaning they can be represented on the number line. These roots can be found where the graph of the function intersects the x-axis, indicating points at which the function evaluates to zero. Understanding real roots is crucial for analyzing the behavior of algebraic functions and their graphs, as they provide insights into the function's value and its transformations.
Reflection: Reflection is a transformation that flips a figure over a specific line, known as the line of reflection, creating a mirror image. This concept is crucial in understanding symmetry and the properties of geometric figures, as it allows for the exploration of how shapes can be altered while maintaining their essential characteristics.
Rolle's Theorem: Rolle's Theorem states that if a function is continuous on a closed interval and differentiable on the open interval between two points, and the function has equal values at these two endpoints, then there exists at least one point in the open interval where the derivative of the function is zero. This theorem is crucial in understanding the behavior of functions and their derivatives, linking concepts like continuity and differentiability to critical points.
Slope: Slope is a measure of the steepness or incline of a line on a graph, defined as the ratio of the vertical change to the horizontal change between two points on that line. It is often represented by the letter 'm' in equations and plays a critical role in understanding how functions behave, analyzing rates of change, and solving optimization problems. Slope can indicate increasing or decreasing trends and can be positive, negative, or zero depending on the orientation of the line.
Translation: Translation refers to the process of shifting a graph of a function in a specific direction without changing its shape or size. This movement is defined by adding or subtracting constants from the function's variables, resulting in a new graph that retains the same characteristics but is positioned differently in the coordinate plane.
Y-intercept: The y-intercept is the point at which a graph crosses the y-axis, indicating the value of the dependent variable when the independent variable is zero. It is a crucial aspect of graphing equations, especially linear functions, as it helps in determining the position of the line in relation to the axes. Understanding the y-intercept can also provide insights into the behavior of more complex functions and their transformations.