# Functions

What is a Function?

Mathematically, a function describes the relationship between two variables. Another way to look at this, is that a function is a rule that gives us a result for any given value of a variable. One can think that, if we enter a number into a function, the function will produce another number. The function changes our original number according to a certain rule.

Example 1

What is the function, or what is the rule by which our number changes, in the table below?

We see that the number seems to double, so the last output should be 2*x*. Our function doubles the number we input.

What is the function, or what is the rule by which our number changes, in the table below?

The function seems to multiply the input by two and then add one to it. So the last output is 2*x* + 1.

The function is usually named by a letter. For example, the letter *f*. The notation *f(x)* means that this is a function *f* with variable *x*. The most commonly used letters for functions are *f*, *g* and *h*. However, it does not matter which letter we use to denote the function. We can even denote *n(x)*, then this is a function *n* and its variable is *x*. The letter we use to represent a function is the name of that function.

**Example 2**

Evaluate the function *f(x) = 2x + 2* with the values 2, 3 and 5 for the variable *x*.

So let's calculate *f(2)*, *f(3)* and *f(5)*.

*f (2) = 2 · 2 + 2 = 6*

*f (3) = 2 · 3 + 2 = 8*

*f (5) = 2 · 5 + 2 = 12*

What is the variable value if the value of function * f(x)* is 0?

*f(x) = 2x + 2*

That is, when *f(x) = 0*.

f (x) = 0

2x + 2 = 0 || −2

2x = −2 || : 2

x = −1

When x = −1 the value of the function is 0. This is the zero of the function.

At what variable value does the value of *f(x) = 2x + 2* is 4?

*f*(*x*) = 4

2x + 2 = 4 || −2

2x = 2 || : 2

*x* = 1

When *x* = 1 the value of the function is 4.

**Example 3**

Below is a graph of a function. Let's find the function value from the graph when *x = 2*.

We go to the point on x-axis where x = 2. We look to where the function graph is at this point and read the value of the function on the y-axis. The value of the function that is 2, *f(2) = 2*

**Turn on the subtitles if needed**