Reference Functions

Introduction

In mathematics, reference functions are fundamental building blocks that help us understand more complex relationships. This chapter explores the following functions:

Square Function: $f (x) = x^{2}$
Square Root Function: $f (x) = \sqrt{x}$
Cube Function: $f (x) = x^{3}$
Inverse Function: $f (x) = \frac{1}{x}$

For each function, we will investigate its definition and equation, its graph and properties and Real-world examples.

A) Square Function

Definition square function

The square function is given by

f (x) = x^{2}

. This means that each input value

x

is multiplied by itself to give the output.

Domain: All real numbers ( $R$ )
Shape: A parabola opening upwards.

Proposition Properties

For any real number $x$ , $x^{2} ⩾ 0$ .
The square function is strictly decreasing on $(- \infty, 0]$ and strictly increasing on $[0, + \infty)$ .
The square function is even (its graph is symmetric with respect to the $y$ -axis).

Proof

Let

f (x) = x^{2}

The product of two real numbers with the same sign is positive, so $x^{2} = x \times x$ is positive.
$\begin{aligned} f (- x) & = (- x)^{2} \\ = (- x) \times (- x) \\ = x^{2} \\ = f (x) . \end{aligned}$ Thus, $f$ is even.
- Let $a$ and $b$ be two real numbers such that $0 ⩽ a < b$ . $\begin{aligned} f (b) - f (a) & = b^{2} - a^{2} \\ = (b - a) (b + a) \\ > 0 & since (b - a) > 0 and (b + a) > 0. \end{aligned}$ Thus, $f (a) < f (b)$ . Therefore, $f$ is strictly increasing on $[0, + \infty)$ .
- Let $a$ and $b$ be two real numbers such that $a < b ⩽ 0$ . $\begin{aligned} f (b) - f (a) & = b^{2} - a^{2} \\ = (b - a) (b + a) \\ < 0 & car (b - a) > 0 and (b + a) < 0. \end{aligned}$ Thus, $f (a) > f (b)$ . Therefore, $f$ is strictly decreasing on $(- \infty, 0]$ .

Example

A square has a side length

x

meters. Its area is given by

A (x) = x^{2}

For example, for

x = 4

meters,

A (4) = 4^{2} = 16

square meters.

B) Square Root Function

Definition Square Root Function

The square root function is given by

f (x) = \sqrt{x}

. It is the inverse of the square function, where the output is the non-negative value that, when squared, gives the input.

Domain: $[0, + \infty)$ (non-negative real numbers)
Shape: A curve that increases rapidly for small $x$ and more slowly as $x$ grows.

Proposition Properties

The square root function is strictly increasing on

[0, + \infty)

Proof

Let

f (x) = \sqrt{x}

.
Let

a

and

b

be two real numbers such that

0 ⩽ a < b

\begin{aligned} f (b) - f (a) & = \sqrt{b} - \sqrt{a} \\ = \frac{(\sqrt{b} - \sqrt{a}) (\sqrt{a} + \sqrt{b})}{\sqrt{a} + \sqrt{b}} \\ = \frac{{(\sqrt{b})}^{2} - {(\sqrt{a})}^{2}}{\sqrt{a} + \sqrt{b}} \\ = \frac{b - a}{\sqrt{a} + \sqrt{b}} \\ > 0 & since b - a > 0 and \sqrt{a} + \sqrt{b} > 0. \end{aligned}

Thus,

f (a) < f (b)

.
The square root function is strictly increasing on

[0, + \infty)

Example

Let a square have an area of

x

square meters. The length of the side of the square is

l (x) = \sqrt{x}

For example, for

x = 25

square meters,

l (x) = \sqrt{25} = 5

meters.

C) Cube Function

Definition Cube Function

The cube function is given by

f (x) = x^{3}

Domain: All real numbers (

R

)

Proposition Properties

The cube function is strictly increasing on $(- \infty, + \infty)$ .
The cube function is odd (its graph is symmetric symmetric about the origin).

Proof

Let

f (x) = x^{3}

$\begin{aligned} f (- x) & = (- x)^{3} \\ = (- x) \times (- x) \times (- x) \\ = - (x \times x \times x) \\ = - x^{3} \\ = - f (x) . \end{aligned}$ Thus, $f$ is odd.
The proof of the variation of the cube function will be done using the derivative tool.

Example

The volume of a cube with a side length

x

meters is

V (x) = x^{3}

For example, for

x = 3

meters,

V (3) = 3^{3} = 27

cubic meters.

D) Inverse Function

Definition Inverse Function

The inverse Function is given by

f (x) = \frac{1}{x}

. It represents a reciprocal relationship, where the output is the reciprocal of the input.

Domain: $R^{*}$ ( $x \neq 0$ )
Shape: a hyperbola.

Proposition Properties

The inverse function is strictly decreasing on $(- \infty, 0)$ and strictly decreasing on $(0, + \infty)$ .
The inverse function is odd (its graph is symmetric symmetric about the origin).

Proof

Let

f (x) = \frac{1}{x}

$\begin{aligned} f (- x) & = \frac{1}{- x} \\ = - \frac{1}{x} \\ = - f (x) . \end{aligned}$ Thus, $f$ is odd.
- Let $a$ and $b$ be two real numbers such that $0 < a < b$ . $\begin{aligned} f (b) - f (a) & = \frac{1}{b} - \frac{1}{a} \\ = \frac{a - b}{b a} \\ < 0 & since a - b < 0 and b a > 0. \end{aligned}$ Thus, $f (a) > f (b)$ .
  Therefore, $f$ is strictly decreasing on $(0, + \infty)$ .
- Similarly, the proof is identical to show that $f$ is strictly decreasing on $(- \infty, 0)$ .