The function, is the most fundamental concept in Mathematics, coined by Leibniz in 1673.

Traditionally, functions define the relation between two variables: every value of $x$ is associated with some value of $y$ .

Additionally, letters from the beginning of the alphabet tend to denote constant values, whereas letters from the end of the alphabet denote variables.

Thoughts: Some may argue that the most fundamental concept in mathematics is the set, the discrete collection of elements, especially given that functions are mappings between sets, sets are required for the concept of the function to be applied- the domain of a function is simply the set of acceptable inputs to it.

A somewhat complicated example of this is the relation between the side of some square and its diagonal.

To express a diagonal as a function of the length of a side, let $x$ be the length of the side, let $y$ be the length of the diagonal, and then we have:

$\Large y = \sqrt{2x^2}$ : $y$ will vary as $x$ does, with $2x^2$ being the operation that associates various values of $y$ , with various values of $x$ .

Commonly, functions are denoted by $f(x)$ , and we can associate the function with some variable representing it's value for some input like so: $y = f(x)$ .

Thus if $y = f(x)$ , and $f(x) = x^2$ , then y is the value of $x^2$ . What's important to note here is that y is the dependent variable, and x is the independent variable.

Explicit & Implicit functions

Rather than say $y = 2x - 7$ , we say $y = f(x) = 2x - 7$ , which means that y, as a function of x, depends on the value of x in this equation: this expression is an explicit function of $x$ .

If the equation were of the equivalent form $2x - y - 7 = 0$ , it is considered an implicit, function of $x$ , because the explicit form is implied by the equation.

Worked Example:

We start with $y = 2x - 7$ (explicit), and choose a number to substitute in for x, say, 43.

Now we have $y = 2(43) - 7$ .

We can simplify $y = 2(43) -> 86, and\; from - 7, we\; get\; 79$ .

Taking the implicit form we get: $2x - y - 7 = 0$ .

Taking the above $x$ value, we get: $2(43) - 79 - 7 =0$ .

We know from before, $2(43) -> 86$ , and we know $86 - 7 = 79$ , so we have: $86 - 79 - 7 = 0, -> 7 - 7 = 0, 0 = 0$ .

Functions can have multiple variables, for example, calculating the hypotenuse of a right triangle, requires two variables, the two sides of the triangle.

For example, $z = f(a, b) = \sqrt{x^2 + y^2}$ .

Physics Lournal

What Is a Function?

The function, is the most fundamental concept in Mathematics, coined by Leibniz in 1673.

Traditionally, functions define the relation between two variables: every value of $x$ is associated with some value of $y$ .

Additionally, letters from the beginning of the alphabet tend to denote constant values, whereas letters from the end of the alphabet denote variables.

A somewhat complicated example of this is the relation between the side of some square and its diagonal.

To express a diagonal as a function of the length of a side, let $x$ be the length of the side, let $y$ be the length of the diagonal, and then we have:

Commonly, functions are denoted by $f(x)$ , and we can associate the function with some variable representing it's value for some input like so: $y = f(x)$ .

Thus if $y = f(x)$ , and $f(x) = x^2$ , then y is the value of $x^2$ . What's important to note here is that y is the dependent variable, and x is the independent variable.

Explicit & Implicit functions

Rather than say $y = 2x - 7$ , we say $y = f(x) = 2x - 7$ , which means that y, as a function of x, depends on the value of x in this equation: this expression is an explicit function of $x$ .

If the equation were of the equivalent form $2x - y - 7 = 0$ , it is considered an implicit, function of $x$ , because the explicit form is implied by the equation.

Worked Example:

We start with $y = 2x - 7$ (explicit), and choose a number to substitute in for x, say, 43.

Now we have $y = 2(43) - 7$ .

We can simplify $y = 2(43) -> 86, and\; from - 7, we\; get\; 79$ .

Taking the implicit form we get: $2x - y - 7 = 0$ .

Taking the above $x$ value, we get: $2(43) - 79 - 7 =0$ .

We know from before, $2(43) -> 86$ , and we know $86 - 7 = 79$ , so we have: $86 - 79 - 7 = 0, -> 7 - 7 = 0, 0 = 0$ .

Functions can have multiple variables, for example, calculating the hypotenuse of a right triangle, requires two variables, the two sides of the triangle.

For example, $z = f(a, b) = \sqrt{x^2 + y^2}$ .

Referenced in

Preliminary Chapters

What Is a Function?

What Is a Function?

The function, is the most fundamental concept in Mathematics, coined by Leibniz in 1673.

Traditionally, functions define the relation between two variables: every value of xxx is associated with some value of yyy.

Additionally, letters from the beginning of the alphabet tend to denote constant values, whereas letters from the end of the alphabet denote variables.

A somewhat complicated example of this is the relation between the side of some square and its diagonal.

To express a diagonal as a function of the length of a side, let xxx be the length of the side, let yyy be the length of the diagonal, and then we have:

Commonly, functions are denoted by f(x)f(x)f(x), and we can associate the function with some variable representing it's value for some input like so: y=f(x)y = f(x)y=f(x).

Thus if y=f(x)y = f(x)y=f(x), and f(x)=x2f(x) = x^2f(x)=x2, then y is the value of x2x^2x2. What's important to note here is that y is the dependent variable, and x is the independent variable.

Explicit & Implicit functions

Rather than say y=2x−7y = 2x - 7y=2x−7, we say y=f(x)=2x−7y = f(x) = 2x - 7y=f(x)=2x−7, which means that y, as a function of x, depends on the value of x in this equation: this expression is an explicit function of xxx.

If the equation were of the equivalent form 2x−y−7=02x - y - 7 = 02x−y−7=0, it is considered an implicit, function of xxx, because the explicit form is implied by the equation.

Worked Example:

We start with y=2x−7y = 2x - 7y=2x−7 (explicit), and choose a number to substitute in for x, say, 43.

Now we have y=2(43)−7y = 2(43) - 7y=2(43)−7.

We can simplify y=2(43)−>86,and from−7,we get 79y = 2(43) -> 86, and\; from - 7, we\; get\; 79y=2(43)−>86,andfrom−7,weget79.

Taking the implicit form we get: 2x−y−7=02x - y - 7 = 02x−y−7=0.

Taking the above xxx value, we get: 2(43)−79−7=02(43) - 79 - 7 =02(43)−79−7=0.

We know from before, 2(43)−>862(43) -> 862(43)−>86, and we know 86−7=7986 - 7 = 7986−7=79, so we have: 86−79−7=0,−>7−7=0,0=086 - 79 - 7 = 0, -> 7 - 7 = 0, 0 = 086−79−7=0,−>7−7=0,0=0.

Functions can have multiple variables, for example, calculating the hypotenuse of a right triangle, requires two variables, the two sides of the triangle.

For example, z=f(a,b)=x2+y2z = f(a, b) = \sqrt{x^2 + y^2}z=f(a,b)=x2+y2​.

Referenced in

Preliminary Chapters

What Is a Function?

Traditionally, functions define the relation between two variables: every value of $x$ is associated with some value of $y$ .

To express a diagonal as a function of the length of a side, let $x$ be the length of the side, let $y$ be the length of the diagonal, and then we have:

Commonly, functions are denoted by $f(x)$ , and we can associate the function with some variable representing it's value for some input like so: $y = f(x)$ .

Thus if $y = f(x)$ , and $f(x) = x^2$ , then y is the value of $x^2$ . What's important to note here is that y is the dependent variable, and x is the independent variable.

Rather than say $y = 2x - 7$ , we say $y = f(x) = 2x - 7$ , which means that y, as a function of x, depends on the value of x in this equation: this expression is an explicit function of $x$ .

If the equation were of the equivalent form $2x - y - 7 = 0$ , it is considered an implicit, function of $x$ , because the explicit form is implied by the equation.

We start with $y = 2x - 7$ (explicit), and choose a number to substitute in for x, say, 43.

Now we have $y = 2(43) - 7$ .

We can simplify $y = 2(43) -> 86, and\; from - 7, we\; get\; 79$ .

Taking the implicit form we get: $2x - y - 7 = 0$ .

Taking the above $x$ value, we get: $2(43) - 79 - 7 =0$ .

We know from before, $2(43) -> 86$ , and we know $86 - 7 = 79$ , so we have: $86 - 79 - 7 = 0, -> 7 - 7 = 0, 0 = 0$ .

For example, $z = f(a, b) = \sqrt{x^2 + y^2}$ .