What Are Functions?

What Are Functions?

Functions are the part of calculus in mathematics. Before begin with indepth learning about function, let us clear our concepts of range, domain, and codomain. 

Some of the most important aspects of a function include Domain, Range, and Codomain. These three basic concepts are the foundation for defining any function. In fact, these may as well make or break an entire function. Hence it is important to have knowledge about each.

Domain is defined as the set of all the values that the function can input while it can be defined.

Range are all the values that come out as output of the function involved.

Codomain is the set of values that have the potential of coming out as outputs of a function.

Intro figure

In this mini lesson, we will explore the world of functions. We will walk through the answers to the questions like what are functios, how to represent a function, what are the conditions for a function, how to solve a function, types of function, and various graphs of functions along with solved examples and interactive questions.

Lesson Plan 


What Are Functions?

In mathematics, a function means a correspondence from one value x of the first set to another value y of the second set. This correspondence can be of the following four types. But each correspondence is not a function.

function means a correspondence from one value x

In the above figure, only 1 – 1 and many to one are examples of a function because no two ordered pairs have the same first component and all elements of the first set are linked in them.

function one input can result in only one output

So we say that in a function one input can result in only one output. If we are given any x then there is one and only one y that can be paired with that x.
The following diagram depicts a function:

diagram depicts a function

A function is a specific type of relation. Any relation may have more than one output for any given input. For example:-

  1. The temperature on any day in a particular city.
  2. The number of calories based on your food intake.
  3. The number of places you can visit with one gallon in your car's fuel tank
  4. The number of sodas coming out of a vending machine depending on how much money you insert.
  5. The amount of carbon left in a fossil after a certain number of years.
  6. The height of a person at a specific age.

But for a function, every x in the first set should be linked to a unique y in the second set. So examples 1, 2, and 3 above are not functions. And examples 4, 5, and 6 are functions.

Representation of Functions

"f(x) = x2 " is the general manner to represent  function.

It is said as f of x is equal to x square.

Special Rules:

  • Function must work for all the possible input value
  • With each input value it must have one relationship only.

How To Solve a Function?

Let us solve an example to understand how to solve a function.

Given two functions: \( \text f = {(-1, 1), (0, 2), (4, 5)}\) and \( \text g = {(1, 1), (2, 3), (7, 9)}\), find \( ( \text g \circ \text f)\) and determine its domain and range.

Here,

\( \begin{align*} \text g ( \text f ( x))  &= \text g \circ \text f \\ &= (\text g \circ \text f)( -1 ) \\ &= \text g [ \text f (-1) ] \\ &= \text g(1) \\ &= 1 \end{align*}\)

Similarly, 

\( \begin{align*} \text g ( \text f ( x))  &= \text g \circ \text f \\ &= (\text g \circ \text f)( 0 ) \\ &= \text g [ \text f (0) ] \\ &= \text g(2) \\ &= 3 \end{align*}\)

Also,

\( \begin{align*} \text g ( \text f ( x))  &= \text g \circ \text f \\ &= (\text g \circ \text f)( 4 ) \\ &= \text g [ \text f (4) ] \\ &= \text g(5) \\ &= \text {undefined} \end{align*}\)

Hence,

\(( \text g \circ \text f ) = {(-1, 1), (0, 3)}\)

Therefore, Domain: {-1, 0} and Range: {0, 3}

\( \begin{align*} \therefore( \text g \circ \text f ) &= {(-1, 1), (0, 3)} \\ \text {Domain} &= (-1,0) \\ \text {Range} &= (0, 3) \end{align*}  \)


Identify Types of Functions With Graph

The various types of functions are:

Identity Function

Identity function is the type of function which gives the same input as the output. 

It is expressed as, 

\(f(x) = x\), where \(x \in \mathbb{R}\)

For example, \(f(3) = 3\) is an identity function.

The following figure shows the plot of an identity function.

Identity Functions

Constant Function

Constant function is the type of function which gives the same value of output for any given input.

It is expressed as, 

\(f(x) = c\), where \(c\) is a constant

For example, \(f(x) = 2\) is a constant function.

The following figure shows the plot of a constant function

Constant function

Polynomial Function

Polynomial function is a type of function which can be expressed as a polynomial.

It is expressed as,

\(f(x) = a_{n}x^{n} + a_{n-1}x^{n-1} +....... + a_{0}x^{0} \)

For example, \(f(x) = 2x+5\) is a polynomial function.

The following figure shows the plot of a polynomial function

\[f\left( x \right) = {x^3} - 6{x^2} + 11x - 6\]

polynomial function

Note that the scale of the two axes is different.

Quadratic Function

Quadratic function is a type of function which has the highest power 2 in the polynomial function.

It is expressed as,

\(f(x)=a(x-h)^{2}+k\).

For example, \(f(x) = x^2 + 1\) is a quadratic function.

The following figure shows the plot of quadratic function

Quadratic Function

Cubic Function

Cubic function is a type of function which has the highest power 3 in the polynomial function.

It is expressed as,

\(f(x) = a_{1}x^{3} + a_{2}x^{2} + a_{3}x + a_{4}\)

For example, \(f(x) = x^3 + 4\) is a cubic function.

The following figure shows the plot of a cubic function

Cubic Function

Rational Function

Rational function is a type of function which is derived from the ratio of two polynomial functions.

It is expressed as, 

\(f(x) = \frac{P(x)}{Q(x)}\), such that P and Q are polynomial functions of \(x\) and \(Q(x)\neq0\)

For example, \(f(x) = \frac{x^2 + 2x + 1}{x^2 - 4}\) is a rational function.

The following figure shows the plot of a rational function.

Rational Function

Modulus Function

Modulus function is a type of function which gives the absolute value of a number by giving its magnitude.

It is expressed as,

\(f(x) = |x| \)

It can further be defined as: 

\(\begin{equation*}
    f(x) = \begin{cases}
               x               & x  \geq 0\\
               -x               & x < 0\\
           \end{cases}
\end{equation*}\)

The following figure shows the plot of a modulus function

Modulus Function


 
important notes to remember
Important Notes
  • A function means a correspondence from one value x of the first set to another value y of the second set. It relates inputs to outputs
  • "f(x) = x2 " is the general manner to represent  function. It is said as f of x is equal to x square.

  • Special Rules:
    a) Function smust work for all the possible input value
    b) With each input value it must have one relationship only.

Solved Examples

Let us have a look at solved examples on what are functions.

Example 1

 

 

Casey is solving a problem on composite functions that says two functions \( \text f \) and \( \text g \) are given by: 

\( \text f (x) = \sqrt{(x - 2)}\)

\( \text g (x) = \ln{(1 + x ^2)} \)

Can you help her find the composite function \( (\text g \circ \text f )( x )\)?

Solution

\( (\text g \circ \text f )( x ) = \text g ( \text f (x))\)

Since \( \text f (x) = \sqrt{(x - 2)}\), substituting \( x = \text f (x)\), we get, 

\( \begin{align*} \text g ( \text f (x)) &= \ln{( 1+ \text f (x)^2)} \\ &= \ln{( 1 + \sqrt{x - 2} ^2)} \\ &= \ln{1 + ( x - 2 )} \\ &= \ln{ (x - 1)}  \end{align*} \)

\(\therefore\) \( (\text g \circ \text f )( x ) = \ln{ (x - 1)}\)

Example 2

 

 

If a function \( \text g (x) = 9 x + 2 \), can you help Alexa in determining what would be the result of \( (\text f \circ \text g )(x)\)?

Solution

\( \text g (x) = 9x + 2 \)

Substitute \( x = \text g ( x ) = 9x + 2 \)

\( \begin{align*} \text f( \text g(x)) &= 9( 9x + 2) + 2 \\ &= 81x + 18 + 2 \\ &= 81x + 20 \end{align*}\)

\(\therefore (\text f \circ \text g )(x) = 81x + 20 \)
Example 3

 

 

Can you help Sam solve the following problem?

\( \text f\) and \( \text g\) are two functions, both defined on the set of real numbers and k is a constant such that,

\( \begin{align*} \text f (x) &= \text k x − 4 \\ \text g (x) &= \text k x + 6 \end{align*}\)

If \( (\text f \circ \text g )(x) = ( \text g \circ \text f)(x)\) for all values of x, what is the value of k?

Solution

Here,

\( \begin{align*} \text f( \text g( x )) &= (k ( \text k x + 6) - 4) \\ &= \text k^2 x + 6 \ \text - \ 4 \end{align*}\)

Similarly,

\( \begin{align*} \text g( \text f( x )) &= (k ( \text kx - 4) + 6) \\ &= \text k^2x - 4 \text k + 6 \end{align*}\)

Since, 

\( \begin{align*} \text k ^2 x + 6 \text k -4 &=  \text k ^2 x - 4 \text k +6 \\ 6 \text k - 4 &= -4 \text k + 6 \\ 10 \text k &= 10 \\ \text k &= 1 \end{align*}  \)

\(\therefore \text k = 1 \)
Example 4

 

 

Help Anthony to prove \(f(2x) \neq 2f(x)\) if \(f(x) = x^3 + 2\)?

Solution

Anthony knows \(f(x) = x^3 + 2\)

To find \(f(2x)\), he will substitute \(x\) by \(2x\)

He will get, \(f(2x) = (2x)^{3} + 2 = 8x^{3} + 2\)

To find \(2f(x)\), he will multiply \(f(x)\) by 2

He will get, \(2f(x) = 2\times(x^{3} + 2) = 2x^{3} + 4\)

Thus, \(8x^{3} + 2 \neq 2x^{3} + 4\)

\(\therefore\) \(f(2x) \neq 2f(x)\)
Example 5

 

 

Help Jack in determining the following:

a) Vertex of a quadratic function \(f(x)=2(x+3)^{2}-2\)
b) Discriminant of a quadratic function \(f(x)=x^{2}+3x-4\)

Solution

a) Let's express the given quadratic function in the standard function \(f(x)=a(x-h)^{2}+k\).

The function \(f(x)=2(x+3)^{2}-2\) can be written as \(f(x)=2(x-(-3))^{2}+(-2)\)

Here, \(h=-3\) and \(k=-2\)

b) The quadratic function is \(f(x)=x^{2}+3x-4\)

On comparing it with \(f(x)=ax^2+bx+c\), we get \(a=1\), \(b=3\), and \(c=-4\)

\[\begin{align}\text{Discriminant}&=b^2-4ac\\&=3^{2}-4(1)(-4)\\&=9+16\\&=25\end{align}\]

\(\therefore\) a).The vertex of the function is (-3, -2

b). The discriminant is 25


Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 
 
 
Challenge your math skills
Challenging Question
1. Find the function composition \( \text f \circ \text g \circ \text h\) if \(\begin{align} \text f (x) = \frac{1}{x + 3}\end{align}\) , \(\text g (x) = \sqrt{( x + 2)}\) , \(\text { and } \text h (x) = x^2 - 2\)
2. Find the value of 'x' in \(75 = 15 e^{0.75x}\).
3. f is a function defined on \(\left[ { -2,1} \right]\) such that \(f\left( x\right) = \frac{1}{2}{x^2}\). Plot the graph of f, and find its domain and range.

Let's Summarize

The mini-lesson targeted the fascinating concept of functions. The math journey around what are functions starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


FAQ's on Functions

Q1. What are parts of a function?

Some of the most important aspects of a function include Domain, Range, and Codomain. These three basic concepts are the foundation for defining any function. In fact, these may as well make or break an entire function. Hence it is important to have knowledge about each.

Domain is defined as the set of all the values that the function can input while it can be defined.

Range, are all the values that come out as output of the function involved.

Codomain is the set of values that have the potential of coming out as outputs of a function.

Q2. How do you write a function?

"f(x) = x2" is the general manner to represent function. It is said as f of x is equal to x square.

Q3. What is the basic concept of function?

Function must work for all the possible input value and with each input value it must have one relationship only.

Q4. How to do transformations of functions?

Follow the rules of the different transformations to find the transformations of functions.

For example, \(f(x+c)\) horizontally shifts the graph of \(f(x)\) left by \(c\) units.

Q5. How to solve cubic functions using the Factor Theorem?

We will consider how to solve the cubic function of the form \(px^3 + qx^2 + rx + s =0\) where p, q, r and s are constants by using the Factor Theorem and Synthetic Division.

Q6. What is a rational function?

A rational function is an algebraic fraction with numerator and denominator as polynomials and the denominator is not equal to zero. 

The coefficients of the polynomials need not be rational numbers.

Q7. How to find the domain of a rational function?

The domain of a rational function consists of all the real numbers x except those for which the denominator is 0.

To find these x values to be excluded from the domain of a rational function, equate the denominator to zero and solve for x

Q8. How do you prove that a function is onto?

By proving Range equal to codomain.

Q9. What is the general form of a quadratic function?

The general form of a quadratic function is \(f(x)=ax^2+bx+c\), where\(a\ne 0\).

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