Glossary

1.2 Basic Classes of Functions

4 min read•june 24, 2024

Linear functions form the foundation of calculus, with their constant rate of change. , the key feature, measures steepness and direction. Understanding linear functions is crucial for grasping more complex relationships in mathematics.

Polynomial functions expand on linear concepts, introducing higher degrees and varied behaviors. These functions exhibit distinct characteristics based on their and coefficients. Mastering polynomials prepares you for analyzing more intricate mathematical relationships and real-world applications.

Linear Functions

Slope of linear functions

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  • Measures the rate of change or steepness of a
  • Calculated as the change in y-values divided by the change in x-values ΔyΔx\frac{\Delta y}{\Delta x}
  • Remains constant for a
  • Positive indicates an increasing function (line rises from left to right)
  • Negative slope indicates a decreasing function (line falls from left to right)
  • Zero slope represents a horizontal line (no change in y-values)
  • Steeper lines have larger absolute values of slope

Polynomial Functions

Polynomial function characteristics

  • Consist of terms with non-negative integer exponents
  • is the highest exponent of the variable
  • Linear functions are first-degree polynomials (ax+bax + b)
  • Quadratic functions are second-degree polynomials (ax2+bx+cax^2 + bx + c)
  • Odd-degree polynomials:
    • Have at least one real ()
    • Exhibit opposite (x→−∞x \to -\infty, f(x)→±∞f(x) \to \pm\infty and x→∞x \to \infty, f(x)→∓∞f(x) \to \mp\infty)
  • Even-degree polynomials:
    • May not have any real roots
    • Exhibit the same (x→±∞x \to \pm\infty, f(x)→∞f(x) \to \infty for positive or f(x)→−∞f(x) \to -\infty for negative leading coefficient)

Quadratic equations and roots

  • In the form ax2+bx+c=0ax^2 + bx + c = 0, where a≠0a \neq 0
  • Roots are x-values where the function equals zero
  • Roots found using the : x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
  • Roots are x-coordinates of points where the graph crosses the x-axis
  • (b2−4acb^2 - 4ac) determines the nature of the roots:
    • Positive discriminant indicates two distinct real roots
    • Zero discriminant indicates one repeated real root
    • Negative discriminant indicates two complex conjugate roots

Other Functions

Types of algebraic functions

  • Rational functions are ratios of two polynomials
    • May have vertical asymptotes where the denominator equals zero
    • May have horizontal asymptotes as x→±∞x \to \pm\infty
  • Power functions are of the form f(x)=xaf(x) = x^a, where aa is constant
    • Positive aa results in an increasing function
    • Negative aa results in a decreasing function
    • Even aa results in a function symmetric about the y-axis
    • Odd aa results in a function symmetric about the origin
  • Root functions are of the form f(x)=xnf(x) = \sqrt[n]{x}, where nn is a positive integer
    • is all non-negative real numbers
    • Graph is increasing and concave down

Algebraic vs transcendental functions

  • Algebraic functions constructed using finite algebraic operations (addition, subtraction, multiplication, division, roots)
    • Include polynomial, rational, power, and root functions
  • are not algebraic
    • Include exponential, logarithmic, and trigonometric functions
    • Often have unique characteristics (asymptotes, periodicity)

Piecewise function graphing

  • Defined by different expressions for different intervals
  • To graph:
    1. Identify the domain intervals for each piece
    2. Graph each piece on its respective interval
    3. Use open or closed circles to indicate endpoint inclusion or exclusion

Function graph transformations

  • Vertical shifts:
    • f(x)+kf(x) + k shifts the graph up by kk units
    • f(x)−kf(x) - k shifts the graph down by kk units
  • Horizontal shifts:
    • f(x−h)f(x - h) shifts the graph right by hh units
    • f(x+h)f(x + h) shifts the graph left by hh units
  • Vertical stretches and compressions:
    • af(x)af(x) stretches the graph vertically by ∣a∣|a| if ∣a∣>1|a| > 1
    • af(x)af(x) compresses the graph vertically by ∣a∣|a| if 0<∣a∣<10 < |a| < 1
  • Horizontal stretches and compressions:
    • f(bx)f(bx) compresses the graph horizontally by ∣b∣|b| if ∣b∣>1|b| > 1
    • f(bx)f(bx) stretches the graph horizontally by ∣b∣|b| if 0<∣b∣<10 < |b| < 1
  • Reflections:
    • −f(x)-f(x) reflects the graph over the x-axis
    • f(−x)f(-x) reflects the graph over the y-axis

Function Properties and Operations

Domain and range

  • Domain: Set of all possible input values (x-values) for which the function is defined
  • : Set of all possible output values (y-values) that result from the function

Function composition and inverse

  • (f ∘ g)(x) combines two functions by applying one function to the output of another
  • f⁻¹(x) "undoes" the original function, swapping input and output values
    • Not all functions have inverses

Continuity and limits

  • A function is continuous at a point if there are no gaps, jumps, or holes in its graph
  • occurs when a function is not continuous at a point
  • describes the behavior of a function as the input approaches a specific value
    • Helps analyze function behavior near points of discontinuity

Key Terms to Review (65)

Algebraic function: An algebraic function is a type of function defined as the root of a polynomial equation with coefficients in the real or complex numbers. It can be expressed using a finite number of operations involving addition, subtraction, multiplication, division, and taking roots.
Algebraic Function: An algebraic function is a mathematical function that can be expressed using a finite combination of algebraic operations, such as addition, subtraction, multiplication, division, and the raising of variables to powers. These functions are widely used in various mathematical and scientific disciplines to model and analyze relationships between variables.
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.
Constant multiple law for limits: The Constant Multiple Law for limits states that the limit of a constant multiplied by a function is equal to the constant multiplied by the limit of the function. Mathematically, if $\lim_{{x \to c}} f(x) = L$, then $\lim_{{x \to c}} [k \cdot f(x)] = k \cdot L$ where $k$ is a constant.
Continuity: Continuity is a fundamental concept in calculus that describes the smoothness and uninterrupted nature of a function. It is a crucial property that allows for the application of calculus techniques and the study of limits, derivatives, and integrals.
Continuity over an interval: Continuity over an interval means that a function is continuous at every point within a given interval. This implies that the function has no breaks, jumps, or holes in that interval.
Cubic function: A cubic function is a polynomial function of degree three, typically expressed in the form $f(x) = ax^3 + bx^2 + cx + d$. The graph of a cubic function can have up to three real roots and two critical points.
Degree: The degree of a polynomial function is the highest power of the variable in the expression. It determines the general shape and behavior of the graph.
Degree: In the context of functions, the degree of a function refers to the highest power of the independent variable present in the function's equation. This characteristic of a function is crucial in determining its behavior and classification within the basic classes of functions.
Discontinuity: Discontinuity refers to a break or interruption in the continuity of a function, where the function's value is not defined or changes abruptly at a particular point. This concept is crucial in understanding the behavior of functions and their derivatives.
Discriminant: The discriminant is a mathematical expression that determines the nature of the roots of a quadratic equation. It is a value that provides information about the number and type of solutions to a quadratic equation.
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 all the values that can be plugged into the function without causing any undefined behavior.
Domain: The domain of a function refers to the set of all possible input values for which the function is defined. It represents the range of values that the independent variable can take on, and it is a crucial concept in understanding the behavior and properties of functions.
End behavior: End behavior describes the behavior of a function's graph as $x$ approaches positive or negative infinity. It is crucial for understanding how functions behave at extreme values.
End Behavior: End behavior refers to the behavior of a function as the input variable approaches positive or negative infinity. It describes the overall trend and characteristics of a function's values as the independent variable becomes increasingly large or small in magnitude.
Exponential Function: An exponential function is a mathematical function where the variable appears as the exponent. These functions exhibit a characteristic pattern of growth or decay, making them important in various fields of study, including calculus, physics, and finance.
Function Composition: Function composition is the process of combining two or more functions to create a new function. The resulting function represents the combined effect of applying the individual functions in a specific order.
Function Transformation: Function transformation is the process of modifying the shape or characteristics of a function by applying various operations or transformations. This concept is essential in understanding the behavior and properties of different classes of functions, as it allows for the manipulation and analysis of their graphical representations and algebraic expressions.
Horizontal asymptote: A horizontal asymptote of a function is a horizontal line that the graph of the function approaches as x tends to positive or negative infinity. It indicates the behavior of the function at extreme values of x.
Horizontal Asymptote: A horizontal asymptote is a horizontal line that a function's graph approaches as the input variable (usually x) approaches positive or negative infinity. It represents the limiting value that the function approaches but never quite reaches.
Horizontal Compression: Horizontal compression is a transformation of a function where the input values are scaled or compressed along the x-axis, resulting in a narrower or more condensed graph of the function. This compression affects the domain and rate of change of the function, while the range and general shape of the function remain unchanged.
Horizontal Shift: A horizontal shift is a transformation of a function that moves the graph of the function left or right along the x-axis, without changing the shape or orientation of the graph. This type of transformation is often used to model real-world phenomena and can be applied to various basic classes of functions.
Horizontal Stretch: A horizontal stretch is a transformation applied to a function that changes the rate of change or slope of the function along the x-axis. It affects the width or spread of the function's graph without altering its height or vertical position.
Infinite discontinuity: An infinite discontinuity occurs at a point where the function approaches infinity or negative infinity as the input approaches a certain value. This results in an unbounded behavior of the function at that specific point.
Inverse function: An inverse function is a function that reverses the effect of the original function. If $f(x)$ is a function, then its inverse $f^{-1}(x)$ satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.
Inverse Function: An inverse function is a function that reverses the relationship between the input and output of another function. It takes the output of the original function and produces the corresponding input, effectively undoing the original function's operation.
Leading coefficient: The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It determines the polynomial's end behavior and influences its graph's shape.
Limit: In mathematics, the limit of a function is a fundamental concept that describes the behavior of a function as its input approaches a particular value. It is a crucial notion that underpins the foundations of calculus and serves as a building block for understanding more advanced topics in the field.
Linear function: A linear function is a polynomial function of degree one, which can be written in the form $f(x) = mx + b$ where $m$ and $b$ are constants. The graph of a linear function is a straight line.
Linear Function: A linear function is a mathematical function where the relationship between the independent and dependent variables is a straight line. This type of function is characterized by a constant rate of change, known as the slope, and is commonly expressed in the form $y = mx + b$, where $m$ represents the slope and $b$ represents the $y$-intercept.
Logarithmic function: A logarithmic function is the inverse of an exponential function and is typically written as $y = \log_b(x)$, where $b$ is the base. It represents the power to which the base must be raised to obtain a given number.
Logarithmic Function: A logarithmic function is a mathematical function that describes an exponential relationship between two quantities. It is the inverse of an exponential function, allowing for the representation of quantities that grow or decay at a constant rate over time. Logarithmic functions are essential in various fields, including mathematics, science, and engineering, for their ability to model and analyze complex phenomena.
Mathematical models: Mathematical models are representations of real-world phenomena using mathematical expressions and concepts. They help in understanding, explaining, and predicting behaviors within a given context.
Natural exponential function: The natural exponential function is defined as $e^x$, where $e$ is Euler's number, approximately equal to 2.71828. It is a fundamental function in calculus with unique properties related to growth and decay.
Piecewise function: A piecewise function is a function that is defined by multiple sub-functions, each applying to a specific interval or condition within its domain. This type of function allows for different formulas to govern different parts of the input values, making it versatile for modeling real-world scenarios where a single formula might not be adequate. Piecewise functions are essential for understanding how functions behave differently across various segments, and they are often analyzed in terms of their continuity at the boundaries between these segments.
Piecewise-defined function: A piecewise-defined function is a function composed of multiple sub-functions, each defined over a specific interval of the domain. These sub-functions can be different expressions that apply to different parts of the domain.
Point-slope equation: The point-slope equation of a line is given by $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope. It is useful for writing the equation of a line when you know one point and the slope.
Polynomial function: A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. These functions are characterized by terms of the form $a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$, where $a_i$ are constants and $n$ is a non-negative integer.
Polynomial Function: A polynomial function is an algebraic function that can be expressed as the sum of one or more terms, each of which consists of a constant (the coefficient) multiplied by one or more variables raised to a non-negative integer power. Polynomial functions are a fundamental class of functions that are widely used in various areas of mathematics, including calculus, and are essential for understanding the behavior of many real-world phenomena.
Power Function: A power function is a mathematical function where the independent variable is raised to a constant power. These functions are characterized by their ability to model exponential growth or decay patterns and are widely used in various scientific and engineering applications.
Quadratic equation: A quadratic equation is a polynomial equation of degree two, typically expressed in the standard form $$ax^2 + bx + c = 0$$, where $$a$$, $$b$$, and $$c$$ are constants and $$a$$ is not equal to zero. Quadratic equations can represent various real-world phenomena, including projectile motion and areas. The solutions to these equations can be found using methods such as factoring, completing the square, or applying the quadratic formula.
Quadratic Formula: The quadratic formula is a mathematical equation used to solve quadratic equations, which are second-degree polynomial equations in the form of $ax^2 + bx + c = 0$. It provides a systematic way to find the roots or solutions of a quadratic equation.
Quadratic function: A quadratic function is a polynomial function of degree 2, which can be written in the form $f(x) = ax^2 + bx + c$, where $a \neq 0$. The graph of a quadratic function is a parabola that opens upwards if $a > 0$ and downwards if $a < 0$.
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 important in calculus as they exhibit unique characteristics and behaviors that are crucial to understanding concepts like rates of change and optimization.
Range: Range refers to the set of all possible output values (or 'y' values) that a function can produce based on its domain (the set of input values). Understanding the range helps us grasp how a function behaves, what outputs are attainable, and the limitations on those outputs.
Rational function: A rational function is a function that can be expressed as the ratio of two polynomials, $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$. These functions are defined for all real numbers except where the denominator is zero.
Rational Function: A rational function is a function that can be expressed as the ratio of two polynomial functions. It is a fundamental class of functions that are widely studied in calculus and have important applications in various fields of mathematics and science.
Reflection: Reflection is a mathematical transformation that involves mirroring a function or graph across a line or axis. It is a fundamental concept that applies to various topics in calculus, including the study of basic function classes and inverse functions.
Root: In the context of basic classes of functions, a root refers to the value of the independent variable that makes the function equal to zero. Roots are essential in understanding the behavior and properties of various function types, as they provide insights into the points where the function intersects the x-axis.
Root function: A root function is a function that involves the extraction of roots, typically square roots or cube roots, of variables. It is commonly represented as $f(x) = \sqrt[n]{x}$ where $n$ is a positive integer.
Root function: A root function is a type of mathematical function that involves taking the root of a variable, typically expressed in the form $$f(x) = \sqrt[n]{x}$$, where $$n$$ is a positive integer. Root functions are essential because they introduce the concept of inverse operations to polynomial functions and help understand how functions behave, especially as they relate to other basic classes of functions like linear and polynomial functions.
Slope: Slope is a measure of the steepness or incline of a line, defined as the ratio of the vertical change to the horizontal change between two points on the line. It is often represented by the letter $m$ and is calculated using the formula $m = \frac{\Delta y}{\Delta x}$.
Slope: Slope is a measure of the steepness or incline of a line or curve, representing the rate of change between two points. It is a fundamental concept in calculus that underpins the understanding of functions, rates of change, and the shape of graphs.
Slope-intercept form: Slope-intercept form is a way to express the equation of a straight line. It is written as $y = mx + b$, where $m$ represents the slope and $b$ represents the y-intercept.
Standard form of a line: The standard form of a line is an equation of the form $Ax + By = C$ where A, B, and C are integers, and A and B are not both zero. This form allows for easy identification of the x- and y-intercepts.
Transcendental Function: A transcendental function is a function that cannot be expressed as a finite combination of algebraic operations, such as addition, subtraction, multiplication, division, and taking roots and powers. These functions are not solutions to algebraic equations and have unique mathematical properties that set them apart from other classes of functions.
Transcendental functions: Transcendental functions are functions that are not algebraic and cannot be expressed as a finite combination of the basic arithmetic operations (addition, subtraction, multiplication, division) and root extractions. Common examples include exponential, logarithmic, and trigonometric functions.
Transformation of a function: A transformation of a function involves shifting, stretching, compressing, or reflecting its graph. These modifications alter the original function's appearance but not its basic shape.
Trigonometric function: A trigonometric function is a mathematical function that relates the angles of a triangle to the lengths of its sides, commonly used in the study of triangles and periodic phenomena. These functions include sine, cosine, tangent, cosecant, secant, and cotangent, each providing a way to describe the relationships between angles and sides in right triangles. They play a crucial role in various areas such as geometry, physics, engineering, and calculus.
Vertical asymptote: A vertical asymptote is a line $x = a$ where the function $f(x)$ approaches positive or negative infinity as $x$ approaches $a$. Vertical asymptotes occur at values of $x$ that make the denominator of a rational function zero, provided that the numerator does not also become zero at those points.
Vertical Asymptote: A vertical asymptote is a vertical line that a function's graph approaches but never touches. It represents the value of the independent variable where the function becomes undefined or experiences a vertical discontinuity.
Vertical Compression: Vertical compression is a transformation that involves scaling a function vertically, effectively shrinking or expanding the function along the y-axis. This transformation alters the amplitude or range of the function while preserving its general shape and behavior.
Vertical Shift: Vertical shift refers to the displacement of a graph or function along the y-axis, either upwards or downwards, without changing the overall shape or orientation of the graph. This transformation affects the y-values of the function, but not the x-values.
Vertical Stretch: Vertical stretch is a transformation that changes the amplitude or scale of a function along the y-axis. It involves multiplying the function by a constant value, which can either expand or compress the function vertically, effectively changing its range and graphical appearance.
X-intercept: The x-intercept of a function is the point where the graph of the function intersects the x-axis, or the value of x when the function's output is equal to zero. It represents the horizontal location where the function crosses the x-axis.
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