# INTRODUCTION TO FUNCTION

A function is a relation in which each possible input value leads to exactly one output value.

In other words, a function is a relation in which each input corresponds with exactly one output.

The input values make up the domain while the output values make up the range.

Functions can be represented using function notation. A function notation shows the functional relationship between input and output values using symbols and signs.

The most common functional notation is $f(x)$, which is read as "f of x".

Other function notations are $h(x)$, $g(x)$.

## Types Of Function

1. Linear function

This is a polynomial function of degree 1. This means the rate of change is constant. Generally, linear functions are usually in the form

$f(x)=Ax+B$

Where A and B are integers but $a≠0$

There are two things you should remember about linear fractions: First, the rate of change is constant. Secondly, the highest power of $x$ in linear function is one.

To illustrate, let's explore this simplified linear function:

If $f(x)=3x+1$. evaluate $f(1)$, $f(2)$, $f(3)$.

$f(1)=3(1)+1$

$f(1)=3+1$

$f(1)=4$

Now to $f(2)$

$f(2)=3(2)+1$

$f(2)=7$

$f(3)=3(3)+1$

$f(3)=10$

As you can see, the rate of change from $f2$ to $f1$ is $7-4=3$. The rate of change from $f3$ to $f2$ is $10-7=3$, which is the same as the first. So, you can see that the rate of change is constant.

This is a polynomial function of the second degree. Any function with one variable $x$ is called a quadratic function if, and only if, it can be written in the form:

$f(x)=ax^2+bx+c$

Where a, b, c are integers but $a≠0$

If a quadratic equation is set equal to zero, the result is a quadratic equation.

3. Cubic function

If the highest power of a polynomial is 3, then we have a cubic function. More precisely, a cubic function is a polynomial function of a third degree.

Generally, cubic functions are usually in this form:

$f(x)=ax^3+bx^2+cx+d$

Where a, b, c, d are numerical values and $a≠0$

4. Logarithmic function

If you have read my post on logarithm, you might have an idea of what is logarithm function. A logarithmic function is one in which the variable appears in the logarithmic form. Generally, the logarithmic function is the inverse function of the exponential function.

logarithmic functions are usually in the form:

$f(x)=\log_{y}x$

Where y is the base and x is the argument of the function.

5. Exponential function

This is a function where one variable appears as an exponent. The general form of the exponential function is given as:

$f(x)=a^x$

Where $a≠0$

6. Trigonometric function

These are the functions of the angle of a triangle. Trigonometric functions are also called circular functions.

$f(x)=sin x$, $f(x)=cos x$ are some examples of trigonometric functions

7. Rational function

Any function which can be written as a ratio of two polynomial function is called a rational function. Polynomial function are usually in this form:

$f(x)=\frac{A(x)}{B(x)}$

where A and B are non-zero integers

Having looked at the types of function, let now evaluate the function

To evaluate function is to determine the output value for a given input value. Stated differently, evaluating function means we should obtain output value for a given input value.

Example 1

If $f(x)=x^2+4$, Evaluate

i) $f(1)$

ii) $f(2)$

Solution:

i) $f(1)$ means we should substitute $1$ for $x$ in the function, accordingly

$f(1)=(1)^2+4$

$f(1)=1+4$

$f(1)=5$

ii) we follow the same procedure.

$f(2)=(2)^2+4$

$f(2)=4+4$

$f(2)=8$

See this post for a detailed discussion on the evaluation of function.

Now, we moved to the inverse of a function

An inverse function is a fraction that can reverse to another function. An inverse function is usually written a using this notation: $f^{-1}(x)$

For instance, $f(x)=x^2$ can inverse to $\sqrt{x}$

But, how do we inverse function? we take an example to illustrate this.

Example 2

Find the inverse of $f(x)=x^3$

Solution:

The first step is to  substitute $y$ for $f(x)$

$y=x^3$

Next, Switch $x$ and $y$

$x=y^3$

When we add cube root to both sides, we have:

$y=\sqrt[3]{x}$

Finally, we back-substitute $f^{-1}(x)$ for $y$

$f^{-1}(x)=\sqrt[3]{x}$

For a detailed discussion on inverse function, refer to this post.

Finally, we round up with composite function

A composite function is a function that is written inside another function. It simply means the outer function that depends on the inner function.

For example, $f[g(x)]$ is a composite function because the function $g(x)$ is written inside $f(x)$

To evaluate the composite function, we simply use the output value of the inner function as the input value for the outer function. let's take an example.

But before that, here are some posts you should read:

Example 3

Given that $f(x)=x+1$ and $g(x)=x+3$, Evaluate $f[g(1)]$

Solution:

First, we evaluate the inner function g(1)

$g(1)=1+3$

$g(1)=4$

Having gotten the output value of g(1), let's use it as input for f(x)

$f(4)=4+1$

$f(4)=5$

With that, we wrap up this post. In the next post, We will be evaluating function.

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