Algebra Chapter 2 Section 2: Functions
Hello internet, and welcome to the Algebra Lecture Series from The Science of Life. In this entry, we move to the definition of an algebraic function.
I feel it is a good idea to start with the definitions of domain and range. The domain is defined as the set of all possible values for which the independent variable (the input, the x-value) can take up. This is all possible values for the horizontal (x) axis. The range is the set of all possible values for which the dependent variable (the output, the y-value) can take up. This is all possible values for the vertical (y) axis.
Now we move to the concept of a function. A good definition of the function is any equation where each input value has exactly one output value. The technical version of this is "any equation where, for each value of the independent variable (the input), there is exactly one value for the dependent variable (the output)." An even more formal definition relies on the definitions of the words domain and range. A function is a correspondence between the domain and range such that for each value of the domain, there is exactly one value of the range.
The function is typically denoted as f(x), which is the symbolic version of the phrase "a function of the independent variable x". Since I have introduced this symbolism, I must emphasize the fact that $f(x)=y$, since the function yields a value on the dependent variable y.
A function can be either an equation -- $y=mx^{2}$, for example -- or it can be a series of points -- for example, (-1,1), (2, 5) and (4, 9). When it's a series of points, the domain is the set of all of the first numbers of the ordered pairs (-1, 2, and 5) while the range is the set of all of the second numbers of the ordered pairs (1, 5, and 9).
When dealing with functions, we have what is called a vertical line test for equations, which is what is says on the tin. It is using a perfectly vertical line to test whether an equation is a function. How it works is if it is impossible to draw a vertical anywhere on the graph of the equation and have it cross the equation at two or more points (regardless of how far away those two points are), then the equation is a function. Otherwise, it's not a function.
In mathematics, there is the concept of a relation. A relation is a correspondence between the domain and the range such that for each element in the domain, there is one or more elements in the range which corresponds to it. That means that a function is also a relation, but a circle is a relation but not a function.
A common analogy for a function is a number crunching machine where you input one number and out comes another number. This is a poor analogy since it implies that it is a black box where you put in a number, magic happens, and out comes another number and no one knows how. A better analogy is making a chair; the raw materials to make the are the input number, the mechanism for constructing the chair from those materials is the function, and the final product is the output.
That's the end of this section. If you have any questions, please leave them is the comments. Like and share this post if you found it helpful. And until next time, stay curious.
To help get this lecture series come out with higher frequency, please donate to The Science of Life or become a Patreon. This helps keep the information current and allows me to dedicate more time to this project instead of obtaining money through external means.
←Previous Chapter 2 Algebra Other Lecture Series Next→
I feel it is a good idea to start with the definitions of domain and range. The domain is defined as the set of all possible values for which the independent variable (the input, the x-value) can take up. This is all possible values for the horizontal (x) axis. The range is the set of all possible values for which the dependent variable (the output, the y-value) can take up. This is all possible values for the vertical (y) axis.
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| The blue horizontal axis (typically the x-axis) is the domain and the independent variable. The green vertical axis (typically the y-axis) is the range and the dependent variable. |
The function is typically denoted as f(x), which is the symbolic version of the phrase "a function of the independent variable x". Since I have introduced this symbolism, I must emphasize the fact that $f(x)=y$, since the function yields a value on the dependent variable y.
A function can be either an equation -- $y=mx^{2}$, for example -- or it can be a series of points -- for example, (-1,1), (2, 5) and (4, 9). When it's a series of points, the domain is the set of all of the first numbers of the ordered pairs (-1, 2, and 5) while the range is the set of all of the second numbers of the ordered pairs (1, 5, and 9).
When dealing with functions, we have what is called a vertical line test for equations, which is what is says on the tin. It is using a perfectly vertical line to test whether an equation is a function. How it works is if it is impossible to draw a vertical anywhere on the graph of the equation and have it cross the equation at two or more points (regardless of how far away those two points are), then the equation is a function. Otherwise, it's not a function.
In mathematics, there is the concept of a relation. A relation is a correspondence between the domain and the range such that for each element in the domain, there is one or more elements in the range which corresponds to it. That means that a function is also a relation, but a circle is a relation but not a function.
A common analogy for a function is a number crunching machine where you input one number and out comes another number. This is a poor analogy since it implies that it is a black box where you put in a number, magic happens, and out comes another number and no one knows how. A better analogy is making a chair; the raw materials to make the are the input number, the mechanism for constructing the chair from those materials is the function, and the final product is the output.
That's the end of this section. If you have any questions, please leave them is the comments. Like and share this post if you found it helpful. And until next time, stay curious.
To help get this lecture series come out with higher frequency, please donate to The Science of Life or become a Patreon. This helps keep the information current and allows me to dedicate more time to this project instead of obtaining money through external means.
←Previous Chapter 2 Algebra Other Lecture Series Next→
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