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, its graph, its properties, and real-world examples.

Square Function

Definition Square Function

The square function is defined on $\mathbb{R}$ by $f(x) = x^2$. This means that each input value $x$ is multiplied by itself to give the output.

Domain: All real numbers ($\mathbb{R}$).
Shape: A curve called a parabola opening upwards.

Proposition Properties

For any real number $x$, $x^2 \geqslant 0$.
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 ($f(-x) = f(x)$).

Proof

Let $f(x) = x^2$.

The product of two real numbers with the same sign is always non-negative, so $x^2 = x \times x \geqslant 0$.
Parity: $f(-x) = (-x)^2 = (-x) \times (-x) = x^2 = f(x)$. Thus, $f$ is even.
Variations:
- Let $a$ and $b$ be two real numbers such that $0 \leqslant a < b$.$$f(b) - f(a) = b^2 - a^2 = (b - a)(b + a)$$Since $b > a$, then $(b - a) > 0$. Since $a$ and $b$ are non-negative and not both zero, $(b + a) > 0$.
  The product is positive, so $f(b) - f(a) > 0$, meaning $f(a) < f(b)$. $f$ is strictly increasing on $[0, +\infty)$.
- Let $a$ and $b$ be two real numbers such that $a < b \leqslant 0$.
  In this case, $(b - a) > 0$ but $(b + a) < 0$.
  The product is negative, so $f(b) - f(a) < 0$, meaning $f(a) > f(b)$. $f$ is strictly decreasing on $(-\infty, 0]$.

Example

The area of a square with side length $x$ is given by $A(x) = x^2$.

If $x = 4$ m, then $A(4) = 4^2 = 16$ m$^2$.

Square Root Function

Definition Square Root Function

The square root function is defined on $[0, +\infty)$ by $f(x) = \sqrt{x}$. For any non-negative $x$, $\sqrt{x}$ is the unique non-negative number whose square is $x$.

Domain: $[0, +\infty)$ (non-negative real numbers).
Shape: A curve starting at the origin $(0,0)$ and increasing.

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 $0 \leqslant a < b$.$$\begin{aligned}[t]f(b) - f(a) &= \sqrt{b} - \sqrt{a} \\ &= \frac{(\sqrt{b} - \sqrt{a})(\sqrt{b} + \sqrt{a})}{\sqrt{b} + \sqrt{a}} \\ &= \frac{(\sqrt{b})^2 - (\sqrt{a})^2}{\sqrt{b} + \sqrt{a}}\\ &= \frac{b - a}{\sqrt{b} + \sqrt{a}}\end{aligned}$$Since $b > a$, the numerator $(b - a)$ is positive. The denominator is a sum of square roots of two non-negative numbers (not both zero), so it is positive.
Thus, $f(b) - f(a) > 0$, so $f(a) < f(b)$.

Cube Function

Definition Cube Function

The cube function is defined on $\mathbb{R}$ by $f(x) = x^3$.

Proposition Properties

The cube function is strictly increasing on $\mathbb{R}$.
The cube function is odd: its graph is symmetric with respect to the origin $(0,0)$.

Proof

Parity: $f(-x) = (-x)^3 = (-x) \times (-x) \times (-x) = -x^3 = -f(x)$.Thus, $f$ is odd.

Example

The volume of a cube with side length $x$ is $V(x) = x^3$.

Inverse Function

Definition Inverse Function

The inverse function is defined on $\mathbb{R}^*$ (all real numbers except $0$) by $f(x) = \frac{1}{x}$.

Domain: $\mathbb{R} \setminus \{0\}$, also written $(-\infty, 0) \cup (0, +\infty)$.
Shape: A curve called a hyperbola.

Proposition Properties

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

Proof

Let $a$ and $b$ be two real numbers such that $0 < a < b$.$$f(b) - f(a) = \frac{1}{b} - \frac{1}{a} = \frac{a - b}{ab}$$Since $a < b$, then $(a - b) < 0$. Since $a$ and $b$ are positive, $ab > 0$.
The quotient is negative, so $f(b) < f(a)$. The order is reversed; $f$ is strictly decreasing on $(0, +\infty)$.