Name: function operations and domain rules
Uploaded: 2025-11-26T05:44:01.000Z
Duration: 1 min 16 s
Channel: JoVE

Funkcje można łączyć na różne sposoby, aby tworzyć nowe — dodając, odejmując, mnożąc, dzieląc lub komponując ich dane wyjściowe.Aby lepiej zrozumieć te kombinacje, rozważmy przykład pokazujący, jak dane wyjściowe z dwóch funkcji oddziałują na siebie w ramach każdej operacji — dodawania, odejmowania, mnożenia i dzielenia.Pokazuje to, że jeśli zmienia się dziedzina funkcji wejściowej, zmienia się również dziedzina danych wyjściowych.Dziedzina funkcji złożonej zawiera tylko wartości wejściowe prawidłowe dla obu funkcji wejściowych.W przypadku dzielenia wykluczone są wartości, które powodują dzielenie przez zero.Kompozycja to inna metoda, w której dane wyjściowe jednej funkcji stają się danymi wejściowymi innej, tworząc funkcję złożoną.Weźmy pod uwagę falę w wodzie, która rozszerza się na zewnątrz po upuszczeniu kamienia.Jedna funkcja modeluje rosnący promień w czasie, a inna używa tego promienia do obliczenia powierzchni.Razem tworzą one funkcję złożoną, która modeluje, jak obszar zmienia się w czasie.

Łączenie funkcji

Funkcje mo&#380;na &#322;&#261;czy&#263;, tworz&#261;c nowe modele matematyczne opisuj&#261;ce interakcje mi&#281;dzy zmiennymi. Te kombinacje s&#261; kluczowe dla zrozumienia relacji mi&#281;dzy zmieniaj&#261;cymi si&#281; wielko&#347;ciami i powszechnie wyst&#281;puj&#261; w dziedzinach naukowych i in&#380;ynierskich. Metody &#322;&#261;czenia &#8212; dodawanie, odejmowanie, mno&#380;enie, dzielenie i kompozycja &#8212; maj&#261; odr&#281;bne konsekwencje dla dziedziny oraz zachowania funkcji wynikowej.Przy &#322;&#261;czeniu funkcji za pomoc&#261; operacji arytmetycznych, takich jak<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:54.84%;'><img style='aspect-ratio:1920/650;' src='https://cloudfront.jove.com/files/ftp_upload/40051/40051_Equation_1_editor_40051.svg' alt='40051_Equation_1.svg' /></figure>dziedzina funkcji wynikowej jest ograniczona do zbioru argument&#243;w wsp&#243;lnego dla dziedzin funkcji f i g. W przypadku dzielenia nak&#322;ada si&#281; dodatkowe ograniczenia, aby wykluczy&#263; wszystkie x, dla kt&#243;rych g(x) = 0, poniewa&#380; dzielenie przez zero jest niezdefiniowane.Na przyk&#322;ad, je&#347;li f(x) = &#8730;x i g(x) = x &#8722; 2, to (f + g)(x) jest okre&#347;lone dla x &#8805; 0, poniewa&#380; funkcja pierwiastka kwadratowego wymaga nieujemnych argument&#243;w. Jednak (f/g)(x) = (f(x)/g(x)) jest okre&#347;lone tylko dla x &gt; 2, gdzie f(x) przyjmuje warto&#347;ci rzeczywiste i jednocze&#347;nie g(x) &#8800; 0.W kompozycji funkcji wynik jednej funkcji staje si&#281; argumentem drugiej, co zapisuje si&#281; jako (f&#8728;g)(x) = f(g(x)). Dziedzina kompozycji funkcji obejmuje wszystkie x takie, &#380;e x nale&#380;y do dziedziny funkcji g, a g(x) mie&#347;ci si&#281; w dziedzinie funkcji f. Taka konstrukcja modeluje procesy sekwencyjne, w kt&#243;rych jedna wielko&#347;&#263; zale&#380;y od drugiej, a ta z kolei od kolejnej.Dziedzina kompozycji funkcji musi uwzgl&#281;dnia&#263; zar&#243;wno funkcj&#281; wewn&#281;trzn&#261;, jak i zewn&#281;trzn&#261;. Na przyk&#322;ad, je&#347;li f(x) = &#8730;x i g(x) = x^2 &#8722; 9, to ((f&#8728;g)(x)) = &#8730;(x^2 &#8722; 9), co wymaga spe&#322;nienia warunku x^2 &#8722; 9 &#8805; 0, daj&#261;c dziedzin&#281; (&#8722;&#8734;, &#8722;3] &#8746; [3, &#8734;). Ka&#380;dy etap przekszta&#322;cenia musi zwraca&#263; warto&#347;ci dopuszczalne jako argumenty w kolejnym kroku kompozycji.

Funkcje mo&#380;na &#322;&#261;czy&#263;, tworz&#261;c nowe modele matematyczne opisuj&#261;ce interakcje mi&#281;dzy zmiennymi. Te kombinacje s&#261; ...

Education

JoVE Core

Math Fundamentals

Podstawy matematyki

Unit

Jednostka

Functions and Their Graphs

Funkcje i ich wykresy

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

Combining Functions

function operations and domain rules

Functions can be combined to make new mathematical models that describe how variables interact. These combinations help explain relationships between changing quantities and often appear in science and engineering. The main methods are addition, subtraction, multiplication, division, and composition, and each one changes the result in a different way.

When two functions are added, subtracted, multiplied, or divided, the new function can only use inputs that work for both original functions. In other words, the domain of the result is the set of input values common to the domains of f and g. Division has one extra rule. Any input x that makes g(x) = 0 must be excluded, because division by zero is undefined.

A simple example uses f(x) = √x and g(x) = x - 2. For (f + g)(x), the domain is x ≥ 0, because the square root function needs nonnegative inputs. For (f / g)(x), the domain is only x &gt; 2, since both functions must give real values and g(x) cannot equal zero.

Function composition works differently. In composition, the output of one function becomes the input of another, written as (f ∘ g)(x) = f(g(x)). The domain of f ∘ g includes every x that is in the domain of g and also gives an output inside the domain of f. This idea models step-by-step processes, where one quantity depends on another, which then depends on a third.

The composite function must satisfy the rules for both the inner and outer functions. For example, if f(x) = √x and g(x) = x² - 9, then (f ∘ g)(x) = √(x² - 9). This requires x² - 9 ≥ 0, so the domain is (-∞, -3] ∪ [3, ∞). Each step must produce an output that can be used in the next step.

Function Operations and Domain Rules

introduction to functions

Functions in Tables, Graphs, and Formulas

Functions describe input-output relationships that follow a clear rule. In a function, each input matches one corresponding output. This makes functions useful for showing how quantities change and for making predictions in math and science. A parking garage fee system is one common example. The input is the number of non-negative hours a car stays inside. Negative time does not make sense in this setting. The total cost starts with a flat entry charge and then rises by a steady hourly rate,...

types of functions i

Polynomial, Rational, and Radical Functions

Functions describe how one quantity changes in relation to another. They are key math tools for showing relationships between variables. Because functions can model real-world patterns, they are used across many types of problems. Algebraic functions are built from basic operations such as addition, subtraction, multiplication, division, and root extraction. The main algebraic types are polynomial, rational, and radical functions. Each one has its own form and use. Polynomial functions use...

types of functions ii

Sine, Cosine, and Exponential Models

Trigonometric and exponential functions help model two major patterns in science and math: repeating cycles and fast growth or decay. These functions extend basic algebra by describing behavior that changes in ways a straight line cannot. They are useful in many real-world settings, especially in science and engineering. Trigonometric functions such as sine and cosine are best for periodic phenomena, which means patterns that repeat over and over. Their smooth curves continue for all real...

types of functions iii

Logarithmic and Piecewise Functions

Logarithmic and piecewise functions help model real-world situations in different ways. A logarithmic function is the inverse of an exponential function. Its graph rises quickly for small x values and then flattens as x gets larger. This makes logarithms useful when data spans several orders of magnitude. Logarithmic scales compress large ranges into compact numbers. One common example is the pH scale in chemistry. It is written as pH = −log10[H+], where [H+] is the hydrogen ion concentration.

rules for defining functions

How to Tell If a Relation Is a Function

A relation is a function when each input x has exactly one output y. This rule helps students decide whether an equation, graph, or table represents a function. The equation y = 2x + 5 is a function because every x-value gives one unique y-value. But x = y^2 + 1 is not a function of x. When x = 2, for example, the equation gives two possible y-values: y = 1 and y = -1. Graphs can be checked with the vertical line test. If a vertical line crosses a curve more than once, the graph fails the...

piecewise-deﬁned functions

Graphing Piecewise Function Intervals

Piecewise defined functions use different rules over different parts of the domain. Each rule applies to a specific interval, so one function can change its behavior when the input changes. These functions are useful for modeling systems that do not stay the same across all input values. A piecewise function can combine more than one expression and still count as a single function. For example, one rule may be linear for inputs less than or equal to -1. Another rule may be quadratic for values...

graphs of functions

Function Graphs and Their Shapes

Function graphs show how output values change when input values change. Each point on the graph is an ordered pair. The x-coordinate is the independent variable, and it sets the horizontal position. The y-coordinate is the dependent variable, and it sets the vertical position. Linear functions such as y = x make a straight line. This shape shows a constant rate of change. Nonlinear functions behave differently and can curve in more complex ways. Even power functions form U-shaped curves.

decreasing function

How to Identify a Decreasing Function

A decreasing function shows a drop in output as the input increases. If one input value is larger than another, the function gives a smaller output. For an interval I, a function f is decreasing when x1 f(x2) for every pair of values in that interval. On a graph, a decreasing function slopes downward from left to right. The same idea can be checked with rate of change. For discrete points, the average rate of change is the change in output divided by the change in input. When that value is...

increasing function

Intervals Where a Function Rises

A function can rise as its input values increase. This pattern is called an increasing function. On a graph, it appears as a line or curve that slopes upward from left to right. An increasing function follows a clear rule. If x1 is less than x2, then f(x1) is less than f(x2). In other words, larger inputs give larger output values. This idea helps describe growth in situations such as population dynamics, financial investments, and resource consumption. The average rate of change shows how...

transformations of functions i

Shifting Function Graphs on the Plane

Shifting function graphs changes their position on the coordinate plane without changing their overall shape. These transformations help move a graph while keeping its pattern and structure the same. Shifting is one of the most common graph transformations. When a constant is added to or subtracted from the output of a function, the graph shifts vertically. A positive change moves the graph upward. A negative change moves it downward. This is like raising or lowering the height of a telescopic...

transformations of functions ii

Horizontal Shifts in Function Graphs

Horizontal shifts move a function’s graph left or right without changing its shape. In this type of transformation, the input variable x is replaced inside the equation. The graph keeps the same overall structure, but its position on the x-axis changes. A shift to the left happens when x is replaced by x + c, where c is a constant. The outputs then appear after a smaller input value, so the graph moves left by c units. A shift to the right happens when x is replaced by x - c. In that case, the...

transformations of functions iii

Graph Reflections and Stretching

Function graphs can be changed without altering their basic shape. These transformations include reflections, vertical stretching, and horizontal compression. They help show how a function changes on a graph while keeping its fundamental form. A reflection flips a graph across an axis. When every output value is multiplied by negative one, the graph mirrors over the horizontal axis. This reverses peaks and troughs, much like signal inversion in electrical systems, where a waveform is flipped...

combining functions

Functions can be combined to make new mathematical models that describe how variables interact. These combinations help explain relationships between changing quantities and often appear in science and engineering. The main methods are addition, subtraction, multiplication, division, and composition, and each one changes the result in a different way. When two functions are added, subtracted, multiplied, or divided, the new function can only use inputs that work for both original functions. In...

introduction to one-to-one functions

Graphing One-to-One Functions and Inverses

One-to-one functions pair each input with a unique output. That means two different values in the domain do not produce the same value in the range. In notation, this is written as f(x1) ≠ f(x2) whenever x1 ≠ x2. The Horizontal Line Test helps identify a one-to-one function from its graph. If no horizontal line crosses the graph at more than one point, the function is one-to-one. A quadratic function such as f(x) = x2 does not meet this condition over its full domain because x = 1 and x = -1...

key terms and concepts

Learn how adding, subtracting, multiplying, dividing, and composing functions changes the domain, with clear examples using square roots and composite functions.

Function Operations and Domain Rules | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Function Operations and Domain Rules

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