Functions - Revision history

Khanh: /* Composition of functions */

2022-01-15T21:11:27Z

Composition of functions

Khanh: /* Composition of functions */

2022-01-15T21:09:30Z

Composition of functions

Khanh: /* Basic concepts of functions */

2021-11-05T22:00:32Z

Basic concepts of functions

Khanh at 21:53, 5 November 2021

2021-11-05T21:53:39Z

Khanh: /* Polynomials */

2021-11-05T21:49:19Z

Polynomials

Khanh: /* Exponential and Logarithmic */

2021-11-05T21:44:45Z

Exponential and Logarithmic

Khanh: /* Resources */

2021-10-24T22:48:15Z

Resources

Lila: /* Resources */

2021-10-21T18:39:43Z

Resources

Lila at 15:42, 4 October 2021

2021-10-04T15:42:17Z

Lila: /* Basic concepts */

2021-10-04T15:37:06Z

Basic concepts

@@ Line 419: / Line 419: @@
 |}
-(Note that this is ''not'' the same as <math>\mbox{sine}(\mbox{square}(x))=\sin(x^2)</math>).
+(Note that this is ''not'' the same as <math>\mbox{sine}(\mbox{square}(x))=\sin(x^2)</math>.)
 Since the function sine equals <math>\tfrac{1}{2}</math> if <math>x=\frac{\pi}{6}</math> ,

@@ Line 419: / Line 419: @@
 |}
-(Note that this is ''not'' the same as <math>\mbox{sine}(\mbox{square}(x))=\sin(x^2)</math> .)
+(Note that this is ''not'' the same as <math>\mbox{sine}(\mbox{square}(x))=\sin(x^2)</math>).
 Since the function sine equals <math>\tfrac{1}{2}</math> if <math>x=\frac{\pi}{6}</math> ,

@@ Line 62: / Line 62: @@
 ===Example===
-Let function <math>f(x)=5x+\frac{5x}{2}</math> for all <math>x</math>. For what value of <math>x</math> gives <math>f(x)=225\frac{1}{2}</math>?}}
+Let function <math>f(x)=5x+\frac{5x}{2}</math> for all <math>x</math>. For what value of <math>x</math> gives <math>f(x)=225\frac{1}{2}</math>
 In mathematics, it is important to notice any repetition. If something seems to repeat, keep a note of that in the back of your mind somewhere.
@@ Line 201: / Line 201: @@
 #Each function has a set of values, the function's ''domain'', which it can accept as input. Perhaps this set is all positive real numbers; perhaps it is the set {pork, mutton, beef}. This set must be implicitly/explicitly defined in the definition of the function. You cannot feed the function an element that isn't in the domain, as the function is not defined for that input element.
 #Each function has a set of values, the function's ''range'', which it can output. This may be the set of real numbers. It may be the set of positive integers or even the set {0,1}. This set, too, must be implicitly/explicitly defined in the definition of the function.
 Functions are an important foundation of mathematics. This modern interpretation is a result of the hard work of the mathematicians of history. It was not until recently that the definition of the relation was introduced by René Descartes in ''Geometry'' (1637). Nearly a century later, the standard notation (<math>f(x)=y</math>) was first introduced by Leonhard Euler in ''Introductio in Analysin Infinitorum'' and ''Institutiones Calculi Differentialis''. The term function was also a new innovation during Euler's time as well, being introduced Gottfried Wilhelm Leibniz in a 1673 letter "to describe a quantity related to points of a curve, such as a coordinate or curve's slope." Finally, the modern definition of the function being the relation among sets was first introduced in 1908 by Godfrey Harold Hardy where there is a relation between two variables <math>x</math> and <math>y</math> such that "to some values of <math>x</math> at any rate correspond values of <math>y</math>."
 ==Important functions==

@@ Line 129: / Line 129: @@
 The set of elements in <math>B</math> is the '''range''' of the function mapping <math>f:A\to B</math> in the cartesian cross product, whereby the set of all elements <math>a\in A</math> maps to some element <math>b\in B</math> .
-The '''codomain''' is the set <math>B</math>, where it is not necessarily the case that all elements <math>b\in B</math> was mapped by some <math>a\in A</math> .}}
+The '''codomain''' is the set <math>B</math>, where it is not necessarily the case that all elements <math>b\in B</math> was mapped by some <math>a\in A</math>.
 Note that the codomain is not as important as the other two concepts.
@@ Line 201: / Line 201: @@
 #Each function has a set of values, the function's ''domain'', which it can accept as input. Perhaps this set is all positive real numbers; perhaps it is the set {pork, mutton, beef}. This set must be implicitly/explicitly defined in the definition of the function. You cannot feed the function an element that isn't in the domain, as the function is not defined for that input element.
 #Each function has a set of values, the function's ''range'', which it can output. This may be the set of real numbers. It may be the set of positive integers or even the set {0,1}. This set, too, must be implicitly/explicitly defined in the definition of the function.
-Functions are an important foundation of mathematics. This modern interpretation is a result of the hard work of the mathematicians of history. It was not until recently that the definition of the relation was introduced by René Descartes in ''Geometry'' (1637). Nearly a century later, the standard notation (<math>f(x)=y</math>) was first introduced by Leonhard Euler in ''Introductio in Analysin Infinitorum'' and ''Institutiones Calculi Differentialis''. The term function was also a new innovation during Euler's time as well, being introduced Gottfried Wilhelm Leibniz in a 1673 letter "to describe a quantity related to points of a curve, such as a coordinate or curve's slope." Finally, the modern definition of the function being the relation among sets was first introduced in 1908 by Godfrey Harold Hardy where there is a relation between two variables <math>x</math> and <math>y</math> such that "to some values of <math>x</math> at any rate correspond values of <math>y</math>." For the person that wants to learn more about the history of the function, go to [[w:History of the function concept|History of functions]].
+Functions are an important foundation of mathematics. This modern interpretation is a result of the hard work of the mathematicians of history. It was not until recently that the definition of the relation was introduced by René Descartes in ''Geometry'' (1637). Nearly a century later, the standard notation (<math>f(x)=y</math>) was first introduced by Leonhard Euler in ''Introductio in Analysin Infinitorum'' and ''Institutiones Calculi Differentialis''. The term function was also a new innovation during Euler's time as well, being introduced Gottfried Wilhelm Leibniz in a 1673 letter "to describe a quantity related to points of a curve, such as a coordinate or curve's slope." Finally, the modern definition of the function being the relation among sets was first introduced in 1908 by Godfrey Harold Hardy where there is a relation between two variables <math>x</math> and <math>y</math> such that "to some values of <math>x</math> at any rate correspond values of <math>y</math>."
 ==Important functions==
@@ Line 224: / Line 224: @@
 When <math>n=2</math>, the polynomial can be rewritten into<blockquote><math>f(x)=ax^2+bx+c</math>, where <math>a,b,c</math> are constants.</blockquote>The graph of this function is a parabola, like the trajectory of a basketball thrown into the hoop.
-If there are questions about the quadratic formula and other basic polynomial concepts, please review the respective chapters in [[Algebra/Polynomials|Algebra]].
+If there are questions about the quadratic formula and other basic polynomial concepts, please review the respective chapters in Algebra.
 === Trigonometric ===
-Trigonometric functions are also very important because it can connect algebra and geometry. Trigonometric functions are explained in detail [[Calculus/Trigonometry|here]] due to its importance and difficulty.[[File:Logexponential.png|thumb|right|The curve on the left is an exponential function while the curve on the right is a logarithmic one|256x256px]]
+Trigonometric functions are also very important because it can connect algebra and geometry. Trigonometric functions are explained in detail here due to its importance and difficulty.[[File:Logexponential.png|thumb|right|The curve on the left is an exponential function while the curve on the right is a logarithmic one|256x256px]]
 === Exponential and Logarithmic ===
 Exponential and logarithmic functions have a unique identity when calculating the derivatives, so this is a great time to review those functions.

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 To be more concise, it can also be written in the summation form:
-:<math>\sum_{k=0}^n a_k x^k</math>}}
+:<math>\sum_{k=0}^n a_k x^k</math>
 ==== Constant ====

@@ Line 233: / Line 233: @@
 :<math>f(x)=a^x</math>, where <math>a</math> is a constant.
 while the logarithmic function is defined as:
-:<math>f(x)=\log_b x</math>, where <math>b</math> is a constant.}}
+:<math>f(x)=\log_b x</math>, where <math>b</math> is a constant.
 A special number will be frequently seen in those functions: the Euler's constant, also known as the base of the natural logarithm. Notated as <math>e</math>, it is defined as <math>e=\sum_{n=0}^\infty \frac{1}{n!}</math>.

@@ Line 512: / Line 512: @@
 ==Resources==
 * [https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:functions/x2f8bb11595b61c86:evaluating-functions/v/what-is-a-function What is a Function?], Khan Academy
-* [https://www.khanacademy.org/math/algebra2/functions_and_graphs Videos on function and their graphs at Khan Academy]
+* [https://www.khanacademy.org/math/algebra2/functions_and_graphs Function and their graphs], Khan Academy
 * [https://www.cliffsnotes.com/study-guides/calculus/precalculus/functions/relations-vs-functions Relations vs. Functions], Cliff's Notes
 * [https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/determine-whether-a-relation-represents-a-function/ Determining if a Relation is a Function], Lumen Learning

← Older revision		Revision as of 18:39, 21 October 2021
Line 516:		Line 516:
	* [https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/determine-whether-a-relation-represents-a-function/ Determining if a Relation is a Function], Lumen Learning		* [https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/determine-whether-a-relation-represents-a-function/ Determining if a Relation is a Function], Lumen Learning
	* [https://courses.lumenlearning.com/waymakercollegealgebra/chapter/identify-functions-using-graphs/ Identifying Functions Using Graphs], Lumen Learning		* [https://courses.lumenlearning.com/waymakercollegealgebra/chapter/identify-functions-using-graphs/ Identifying Functions Using Graphs], Lumen Learning
		+
		+	== Licensing ==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikibooks.org/wiki/Calculus/Functions Functions, Wikibooks: Calculus] under a CC BY-SA license

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 Again, the above definitions are often very confusing. Again, another image is provided to the right of the bullet points. Along with that, another example is also provided. Let us analyze the function <math>g(x)=ax^{2}</math>.
-{{ExampleRobox|theme=2|title='''Given''': <math>g(x)=ax^{2}</math>, where <math>a</math> is constant and <math>a\ne0</math>.<br>
-'''Prove''': function <math>g</math> is non-surjective and non-injective.}} Notice that the only changing element in the function <math>g</math> is <math>x</math>. Let us check to see the conditions of this function.
+'''Given''': <math>g(x)=ax^{2}</math>, where <math>a</math> is constant and <math>a\ne0</math>.<br>
 '''Is it injective'''? Let <math>g(x)=y</math>, and solve for <math>x</math>. First, divide by <math>a</math>.
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 Notice, however, what we learned from the above manipulation. As a result of solving for <math>x</math>, we found that there are two solutions for <math>x</math>. However, this resulted in two different values from <math>\frac{y}{a}</math>. This means that for some individual <math>x</math> that gives <math>y</math>, there are two different inputs that result in the same value. Because <math>f(x_{1})=f(x_{2})</math> when <math>x_{1}\ne x_{2}</math>, this is by definition non-injective.
-'''Is it surjective'''? As a natural consequence of what we learned about inputs, let us determine the range of the function. After all, the only way to determine if something is surjective is to see if the range applies to all real numbers. A good way to determine this is by finding a pattern involving our domains. Let us say we input a negative number for the domain: <math>f(-x)=a(-x)^{2}=a\cdot(-x)\cdot(-x)=ax^2</math>. This seemingly trivial exercise tells us that negative numbers give us non-negative numbers for our range. This is huge information! For <math>x\in\mathbb{R}</math>, the function always results <math>y>0</math> for our range. For the set <math>B</math>, we have elements in that set that have no mappings from the set <math>A</math>. As such, <math>y\le0</math> is the codomain of set <math>B</math>. Therefore, this function is non-surjective!{{Robox/Close}}
+'''Is it surjective'''? As a natural consequence of what we learned about inputs, let us determine the range of the function. After all, the only way to determine if something is surjective is to see if the range applies to all real numbers. A good way to determine this is by finding a pattern involving our domains. Let us say we input a negative number for the domain: <math>f(-x)=a(-x)^{2}=a\cdot(-x)\cdot(-x)=ax^2</math>. This seemingly trivial exercise tells us that negative numbers give us non-negative numbers for our range. This is huge information! For <math>x\in\mathbb{R}</math>, the function always results <math>y>0</math> for our range. For the set <math>B</math>, we have elements in that set that have no mappings from the set <math>A</math>. As such, <math>y\le0</math> is the codomain of set <math>B</math>. Therefore, this function is non-surjective!
 [[Image:Vertlinetest.png|thumb|This is an example of an expression which fails the vertical line test.]]
@@ Line 201: / Line 204: @@
 ==Important functions==
 === Polynomials ===
 Polynomial functions are the most common and most convenient functions in the world of calculus. Their frequent appearances and their applications on computer calculations have made them important.
-{{Calculus/Def|title=Definition of a polynomial function|text=A polynomial in a single indeterminate x can always be written (or rewritten) in the form:
+A polynomial in a single indeterminate x can always be written (or rewritten) in the form:
 :<math>f(x)=a_n x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\cdot\cdot\cdot+a_2x^2+a_1x+a_0</math>
@@ Line 228: / Line 230: @@
 === Exponential and Logarithmic ===
 Exponential and logarithmic functions have a unique identity when calculating the derivatives, so this is a great time to review those functions.
-{{Calculus/Def|title=Definition for exponential and logarithmic functions|text=The exponential function is defined as:
+The exponential function is defined as:
 :<math>f(x)=a^x</math>, where <math>a</math> is a constant.
 while the logarithmic function is defined as:
@@ Line 307: / Line 309: @@
 <math>\therefore f(x) </math> is an even function
 ==Manipulating functions==
 ===Addition, Subtraction, Multiplication and Division of functions===
 For two real-valued functions, we can add the functions, multiply the functions,
-raised to a power, etc.
+raised to a power, etc. For example:
-If we subtract <math>y=3x+2</math> from <math>y=x^2</math> , we obtain <math>y=x^2-(3x+2)</math> . We can also write this as <math>y=x^2-3x-2</math> .
+* If we add the functions <math>y=3x+2</math> and <math>y=x^2</math> , we obtain <math>y=x^2+3x+2</math> .
-If we multiply the function <math>y=3x+2</math> and the function <math>y=x^2</math> , we obtain <math>y=(3x+2)x^2</math> . We can also write this as <math>y=3x^3+2x^2</math> .
+* If we subtract <math>y=3x+2</math> from <math>y=x^2</math> , we obtain <math>y=x^2-(3x+2)</math> . We can also write this as <math>y=x^2-3x-2</math> .
-If we divide the function <math>y=3x+2</math> by the function <math>y=x^2</math> , we obtain <math>y=\frac{3x+2}{x^2}</math> .
+* If we multiply the function <math>y=3x+2</math> and the function <math>y=x^2</math> , we obtain <math>y=(3x+2)x^2</math> . We can also write this as <math>y=3x^3+2x^2</math> .
 If a math problem wants you to add two functions <math>f</math> and <math>g</math> , there are two ways that the problem will likely be worded:
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 Similar statements can be made for subtraction, multiplication and division.
 Let <math>f(x)=3x+2</math> and: <math>g(x)=x^2</math> . Let's add, subtract, multiply and divide.
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              &= \frac{3}{x}+\frac{2}{x^2}
 \end{align}</math> .
 ===Composition of functions===
 We begin with a fun (and not too complicated) application of composition of functions before we talk about what composition of functions is.
-{{ExampleRobox|theme=2|title=Example: Dropping a ball}}
+Example: Dropping a ball
 If we drop a ball from a bridge which is 20 meters above the ground, then the height of our ball above the earth is a function of time. The physicists tell us that if we measure time in seconds and distance in meters, then the formula for height in terms of time is <math>h=-4.9t^2+20</math> . Suppose we are tracking the ball with a camera and always want the ball to be in the center of our picture. Suppose we have <math>\theta=f(h)</math> The angle will depend upon the height of the ball above the ground and the height above the ground depends upon time. So the angle will depend upon time. This can be written as <math>\theta=f(-4.9t^2+20)</math> . We replace <math>h</math> with what it is equal to. This is the essence of composition.
 Composition of functions is another way to combine functions which is different from addition,
 subtraction, multiplication or division.
 The value of a function <math>f</math> depends upon the value of another variable <math>x</math> ; however, that variable could be equal to another function <math>g</math> , so its value depends on the value of a third variable. If this is the case, then the first variable is a function <math>h</math> of the third variable; this function (<math>h</math>) is called the '''composition''' of the other two functions (<math>f</math> and <math>g</math>).
-{{ExampleRobox|theme=2|title=Example:  Composing two functions}}
+Example:  Composing two functions
 Let <math>f(x)=3x+2</math> and: <math>g(x)=x^2</math> . The composition of <math>f</math> with <math>g</math> is read as either "f composed with g" or "f of g of x."
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 :so <math>f(g(x))\ne g(f(x))</math> .
 Composition of functions is very common, mainly because functions themselves are common. For instance, squaring and sine are both functions:

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 The codomain is all the elements in set <math>B</math>.]]There are a few more important ideas to learn from the modern definition of the function, and it comes from knowing the difference between domain, range, and codomain. We already discussed some of these, yet knowing about sets adds a new definition for each of the following ideas. Therefore, let us discuss these based on these new ideas. Let <math>A</math> and <math>B</math> be a set. If we were to combine these two sets, it would be defined as the ''cartesian cross product'' <math>A\times B</math>. The subset of this product is the function. The below definitions are a little confusing. The best way to interpret these is to draw an image. To the right of these definitions is the image that corresponds to it.
-{{Calculus/Def|title=Definition of domain, range, and codomain of a function|text=The '''domain''' is defined to be a set <math>A</math>
 with all elements <math>a\in A</math> mapping to at least one unique <math>b\in B</math> .