Polynomial, Rational and Irrational Functions

Lessons

Senior 4 · Mathematics

Polynomial, Rational and Irrational Functions

5 lessons

Lesson 4 of 5

Even and odd functions, and rational and irrational models

25 min

Half the work, twice the picture

Fold the graph of $y = x^{2}$ along the $y$ -axis and the two halves match. Turn the graph of $y = x^{3}$ through half a turn about the origin and it lands on itself. Recognising these symmetries from the formula, before plotting a single point, halves the work of drawing a graph or making a table. The second half of this lesson puts rational and irrational functions to work in models from economics and physics.

Definition

A set $D$ is symmetric about 0 if $x \in D \Rightarrow -x \in D$ . Let $f$ have such a domain.

$f$ is even if $f (- x) = f (x)$ for every $x$ in $D$ . Its graph is symmetric about the $y$ -axis.
$f$ is odd if $f (- x) = - f (x)$ for every $x$ in $D$ . Its graph is symmetric about the origin (a half-turn maps it onto itself).

The parity of $f$ is whether it is even, odd or neither.

Example: $x^{2}$ , $x^{4} - 4 x^{2}$ and $∣ x ∣$ are even; $x^{3}$ , $x^{3} - 3 x$ and $\frac{1}{x}$ are odd.

Even: $f (x) = x^{4} - 4 x^{2}$ . The points $(1, - 3)$ and $(- 1, - 3)$ are mirror images in the $y$ -axis.

Odd: $f (x) = x^{3} - 3 x$ . The points $(2, 2)$ and $(- 2, - 2)$ are images of each other under a half-turn about O, which is the midpoint of the dashed segment.

Important

Useful facts about parity

A polynomial is even if it has only even powers of $x$ (a constant counts as $x^{0}$ ), and odd if it has only odd powers.
Even × even and odd × odd are even; even × odd is odd. The same holds for quotients.
If $f$ is odd and $0 \in D$ , then $f (0) = 0$ , since $f (0) = - f (0)$ .
If the domain is not symmetric about 0, the function is neither even nor odd.

Worked example 1: test the parity

Study the parity of (a) $f(x) = \frac{x^3}{x^2 - 4}$ (b) $g(x) = \sqrt{4 - x^2}$ (c) $h (x) = x^{2} + x$ (d) $k(x) = \sqrt{x + 1}$ .

Solution

Domain $\mathbb{R} \setminus \{-2, 2\}$ , symmetric. $f(-x) = \frac{(-x)^3}{(-x)^2 - 4} = \frac{-x^3}{x^2 - 4} = -f(x)$ : odd.
Domain $[- 2, 2]$ , symmetric. $g(-x) = \sqrt{4 - (-x)^2} = g(x)$ : even.
Domain $\mathbb{R}$ . $h (1) = 2$ and $h (- 1) = 0$ : since $h(-1) \ne h(1)$ and $h(-1) \ne -h(1)$ , $h$ is neither.
Domain $[-1, +\infty)$ is not symmetric (3 is in it, $- 3$ is not): neither.

Worked example 2: using parity

(a) $f$ is odd and $f (2) = 5$ . Find $f (- 2)$ and $f (0)$ , given that $f$ is defined at 0. (b) Write $p (x) = x^{3} + 2 x^{2} - x + 5$ as the sum of an even and an odd polynomial.

Solution

$f (- 2) = - f (2) = - 5$ , and $f (0) = 0$ .
Group the even and the odd powers: $p (x) = (2 x^{2} + 5) + (x^{3} - x)$ , with $2 x^{2} + 5$ even and $x^{3} - x$ odd.

Warning

To show that a function is not even or odd, one counter-example is enough (as in part (c)). To show that it is even or odd, you must check $f (- x)$ for a general $x$ : a few numerical values prove nothing.

Rational and irrational functions as models

A rational function is a quotient $\frac{P(x)}{Q(x)}$ of two polynomials; an irrational function contains the variable under a root sign. Both appear wherever a quantity is "shared out" (a division) or found from a square (a root).

Worked example 3: average cost (illustrative figures)

A small workshop pays 200 000 Frw a month in fixed costs, and each chair costs 1 500 Frw in materials. The total cost of $x$ chairs is $C(x) = 200\,000 + 1500x$ , and the average cost per chair is

A(x) = \frac{C(x)}{x} = 1500 + \frac{200\,000}{x}, \qquad x \gt 0 .

(a) Find $A (100)$ and $A (400)$ . (b) How many chairs must be made for the average cost to fall below 2 000 Frw? (c) What happens to $A (x)$ for very large $x$ ?

Solution

$A (100) = 1500 + 2000 = 3500$ Frw; $A (400) = 1500 + 500 = 2000$ Frw.
$1500 + \frac{200\,000}{x} \lt 2000 \iff \frac{200\,000}{x} \lt 500 \iff x \gt 400$ (we may multiply by $x$ because $x \gt 0$ ). More than 400 chairs.
$\frac{200\,000}{x}$ becomes very small, so $A (x)$ gets closer and closer to 1 500 Frw but never reaches it: the fixed costs are shared over more and more chairs.

The average cost falls towards 1 500 Frw (dashed line) but never reaches it. The point $(400, 2000)$ marks the answer to part (b).

Worked example 4: the period of a pendulum

For small swings, the period (time for one full swing, there and back) of a pendulum of length $L$ metres is

T(L) = 2\pi\sqrt{\frac{L}{g}}, \qquad L \gt 0, \quad g = 9.81\,\mathrm{m\,s^{-2}} .

(a) Find $T$ for $L = 1\,\mathrm{m}$ and for Foucault's $L = 67\,\mathrm{m}$ . (b) Find the inverse function $L (T)$ , and the length that gives $T = 2\,\mathrm{s}$ . (c) What happens to $T$ when $L$ is multiplied by 4?

Solution

$T(1) = 2\pi\sqrt{\frac{1}{9.81}} \approx 2.01\,\mathrm{s}$ ; $T(67) = 2\pi\sqrt{\frac{67}{9.81}} \approx 16.4\,\mathrm{s}$ .
$\frac{T}{2\pi} = \sqrt{\frac{L}{g}} \Rightarrow L = \frac{gT^2}{4\pi^2}$ (for $T \gt 0$ ). For $T = 2$ : $L = \frac{9.81 \times 4}{4\pi^2} \approx 0.994\,\mathrm{m}$ .
$T(4L) = 2\pi\sqrt{\frac{4L}{g}} = 2 \times 2\pi\sqrt{\frac{L}{g}} = 2T(L)$ : the period doubles.

Worked example 5: a lens (rational function)

For a converging lens of focal length $f = 10\,\mathrm{cm}$ , an object at distance $u$ gives an image at distance $v$ with $\frac{1}{u} + \frac{1}{v} = \frac{1}{f}$ (real-is-positive convention). Express $v$ as a function of $u$ , find $v$ for $u = 15\,\mathrm{cm}$ , and describe what happens as $u$ approaches 10 cm from above.

Solution $\frac{1}{v} = \frac{1}{10} - \frac{1}{u} = \frac{u - 10}{10u}$ , so $v(u) = \frac{10u}{u - 10}$ for $u \gt 10$ . Then $v(15) = \frac{150}{5} = 30\,\mathrm{cm}$ . As $u$ gets close to 10, the denominator $u - 10$ gets close to 0 while the numerator is close to 100, so $v$ becomes very large: the image moves far away. Unit 7 describes this with limits.

Tip

In a model, the domain comes from the situation as well as from the formula: a number of chairs is a positive whole number, a length is positive, and a real image needs $u \gt f$ . State these conditions before you calculate.

The Organic Chemistry Tutor: Even, odd or neither functions (about 18 min). Watch how f(−x) is simplified and compared with f(x) and with −f(x), and how the result shows up as symmetry in the graph.

✦

Practice

For questions 1 to 4, justify your answer algebraically.

Study the parity of $f (x) = 3 x^{4} - x^{2} + 7$ and $g (x) = 2 x^{5} - x$ .
Study the parity of $f(x) = \frac{x^2 + 1}{x}$ .
Study the parity of $f(x) = \frac{x}{x - 1}$ .
Show that $f(x) = x\sqrt{9 - x^2}$ is odd.
$g$ is even, and $g (3) = - 2$ . Find $g (- 3)$ . Can you find $g (0)$ ?
With the chair workshop above, how many chairs give an average cost of 1 750 Frw?
Find the length of a pendulum whose period is 1 s (take $g = 9.81\,\mathrm{m\,s^{-2}}$ ).
For the lens with $f = 10\,\mathrm{cm}$ , find $u$ when $v = 25\,\mathrm{cm}$ .

Show answersHide answers

$f$ : only even powers, $f (- x) = f (x)$ , even. $g$ : only odd powers, $g (- x) = - g (x)$ , odd.
Domain $\mathbb{R} \setminus \{0\}$ , symmetric; $f(-x) = \frac{x^2 + 1}{-x} = -f(x)$ : odd.
Domain $\mathbb{R} \setminus \{1\}$ is not symmetric ( $- 1$ is in it, $1$ is not): neither.
Domain $[- 3, 3]$ ; $f(-x) = -x\sqrt{9 - x^2} = -f(x)$ .
$g (- 3) = g (3) = - 2$ . No: an even function can take any value at 0.
$\frac{200\,000}{x} = 250 \Rightarrow x = 800$ chairs.
$L = \frac{9.81 \times 1^2}{4\pi^2} \approx 0.248\,\mathrm{m}$ , about 25 cm.
$\frac{1}{u} = \frac{1}{10} - \frac{1}{25} = \frac{3}{50}$ , so $u = \frac{50}{3} \approx 16.7\,\mathrm{cm}$ .

Quick check

Match each function with its parity.

Tap one on the left, then its partner on the right.

Key points

Even: $f (- x) = f (x)$ , symmetric about the $y$ -axis. Odd: $f (- x) = - f (x)$ , symmetric about the origin. Both need a domain symmetric about 0.
One counter-example shows "not even" or "not odd"; a proof needs a general $x$ .
Rational models share something out (average cost, lens images); irrational models contain a square root (pendulum period). The situation adds conditions to the domain.

Sources and credits

Written by Ganzaa: the explanations, the graphs, the worked examples (the workshop's figures are illustrative), the practice items and the quiz questions. The unit follows the topic list of REB's S4 Core Mathematics (2026–2027); no REB text is reproduced.
Photo: Foucault pendulum at Panthéon de Paris, August 2023, Benoît Prieur, Wikimedia Commons, CC0. The bob's mass and the wire's length are from the Wikipedia article Foucault pendulum (CC BY-SA 4.0).
Video: Even, Odd, or Neither Functions The Easy Way!, The Organic Chemistry Tutor, embedded from YouTube.
This lesson adapts CC BY-SA material and is shared under CC BY-SA 4.0.

1Half the work, twice the picture

2Rational and irrational functions as models

Half the work, twice the picture

Rational and irrational functions as models