Notation of functions is the way in which a function is written. Notation of functions is an accurate way of giving information about the function without the need for lengthy written explanations.

In this article, we will make a brief revision of functions. Then, we will look at their notation along with some examples.

## What is a function?

In mathematics, a function is a relation with a set of inputs that have only one output in each case. All functions have a domain and a range. The domain is the set of independent values of the variable, for which the function is defined. That is, the domain is the set of values of *x* for which there are real values of *y*.

On the other hand, the range is the set of dependent values of the variable, *y*. Both range and domain can be expressed in interval notation or using inequalities.

## What is function notation?

Function notation is a way in which a function can be represented using symbols and signs. Function notation is a simpler way to write functions without the need to write extensive written explanations.

The most frequently used function notation is $latex f(x)$, which is read as “*f* of *x*“. In this case, the letter *x* inside the parentheses represents the domain of the function, and the integer symbol $latex f(x)$ represents the range of the function.

Although *f* is the most frequently used letter, we can use any other letter of the alphabet, both lowercase, and uppercase. Similarly, we can also use any other letter of the alphabet instead of *x*.

### EXAMPLES

All of the following are functions:

- $latex f(x)=x-21$
- $latex h(x)={{x}^{2}}+2$
- $latex S(t)=3{{t}^{2}}-t+3$
- $latex jhon(b)={{b}^{3}}-2b$

## Advantages of using function notation

**•** This notation allows us to give individual names to functions and avoid confusion when evaluating them. For example, by having $latex f(x)$ and $latex g(x)$, we can easily distinguish them.

**• **The independent variable can be easily identified. For example, in the function $latex f(x)=2x+3$, we know that the variable is *x*.

**• **We can determine which element of the function should be examined. For example, finding $latex f(3)$ when $latex f(x)=4x-2$ is the same as finding $latex y=4x-2$ when $latex x=3$.

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## Types of functions

There are several types of functions, but the following are the most common:

### Linear functions

A linear function is a first-degree polynomial. A linear function has the general form $latex f(x)=ax+b$, where a and b are numerical values and a is nonzero.

### Quadratic functions

A quadratic function is a second-degree polynomial function. The general form of a quadratic function is $latex f(x)=a{{x}^2}+bx+c$, where *a*, *b*, and *c* are numerical values and *a* is nonzero.

### Cubic function

The cubic function is a third-degree polynomial function that has the general form $latex f(x)=a{{x}^3}+b{{x}^2}+cx+d$, where *a*, *b*, *c* and *d* are numerical values and are different from zero.

### Trigonometric function

The three fundamental trigonometric functions are $latex f(x)=\sin(x)$, $latex f(x)=\cos(x)$, $latex f(x)=\tan(x)$.

### Exponential function

An exponential function is a function in which the variable appears as an exponent. It has the general form $latex f(x)={{b}^x}$.

### Logarithmic function

A logarithmic function is a function in which the variable appears as the argument of a logarithm. It has the general form $latex \log_{b}x$, where *b* is the base of the logarithm.

## Function notation examples

We can apply function notation to evaluate math problems more easily as shown in the examples:

### EXAMPLE 1

Given the function $latex f(x)=3x-5$, find the value of $latex f(3)$.

**Solution:** We substitute the 3 instead of the *x* in the function:

$latex f(x)=3x-5$

$latex f(3)=3(3)-5$

$latex f(3)=9-5$

$latex f(3)=4$

This answer can be thought of as the ordered pair (3, 4).

### EXAMPLE 2

Find the value of the function $latex g(t)=2{{t}^{2}}-t+3$, when $latex t=-2$.

**Solution:** We substitute the 3 instead of the *x* in the function:

$latex g(t)=2{{t}^{2}}-t+3$

$latex g(-2)=2{{(-2)}^{2}}-(-2)+3$

$latex g(t)=2(4)+2+3$

$latex g(t)=13$

This answer can be thought of as the ordered pair (-2, 13).

### EXAMPLE 3

Find the value of $latex h(2s)$, for the function $latex h(x)=3x-16$.

**Solution:** This is similar to the previous problems with the difference that we now substitute variables:

$latex h(x)=3x-16$

$latex h(2s)=3(2s)-16$

$latex h(2s)=6s-16$

Now, the function is in terms of *s*.

### EXAMPLE 4

Given the function $latex g(s)=3{{s}^{3}}+2s-10$, find the value of $latex g(3)$.

**Solution:** We just have to substitute the 3 instead of the *s* in the function:

$latex g(s)=3{{s}^{3}}+2s-10$

$latex g(3)=3{{(3)}^{3}}+2(3)-10$

$latex g(3)=3(27)+6-10$

$latex g(3)=81+6-10$

$latex g(3)=77$

We can present this answer as the ordered pair (3, 77).

### EXAMPLE 5

Given the function $latex f(x)=2{{x}^{2}}+x-5$, find the value of $latex f(m+1)$.

**Solution:** Instead of replacing a number, we replace with $latex m+1$:

$latex f(x)=2{{x}^{2}}+x-5$

$$f(m+1)=2{{(m+1)}^{2}}+m+1-5$$

$$f(m+1)=2({{m}^{2}}+2m+1)+m-4$$

$$f(m+1)=2{{m}^{2}}+4m+2+m-1$$

$latex f(m+1)=2{{m}^{2}}+5m+1$

Now, we have a function in terms of *m.*

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## See also

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