MYP Extended Mathematics Topic Rational Functions…

Rational Functions and Restrictions

Rational functions are a powerful way to model relationships where one quantity depends on the reciprocal of another (for example, "as input gets close to a certain value, output grows without bound"). In many applications this kind of behaviour appears naturally, and graphs help you see it immediately through asymptotes.

Note

Definition: Rational Function

A rational function is any function that can be written as a quotient $\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x)\neq 0$.

The condition $Q(x)\neq 0$ is not a minor detail. It creates restrictions on the domain, and those restrictions often show up on the graph as vertical asymptotes (or sometimes holes, in more advanced cases).

A key "parent" example is the reciprocal function:

Note

Definition: Reciprocal Function

The reciprocal function is $f(x)=\frac{1}{x}$ with $x\neq 0$.

Its domain excludes 0 because division by 0 is undefined.

Note

In many textbooks, the word "reciprocal" refers to the multiplicative inverse of a number (for example, the reciprocal of 5 is $\frac{1}{5}$). The reciprocal function $y=\frac{1}{x}$ turns each input $x$ into its multiplicative inverse, but that is different from the idea of an inverse function (which reverses a whole input-output process).

Properties of the Reciprocal Function

The graph of $y=\frac{1}{x}$ has two separate curves (often called branches): one in Quadrant I and one in Quadrant III.

Behaviour Near Zero

As $x$ gets closer to 0:

as $x\to 0^+$, $\frac{1}{x}\to +\infty$
as $x\to 0^-$, $\frac{1}{x}\to -\infty$

So the graph gets closer and closer to the vertical line $x=0$ but never touches it.

Note

Definition: Vertical Asymptote

A vertical asymptote is a vertical line $x=c$ that a graph approaches as the input approaches $c$. For rational functions, it often occurs where the denominator approaches 0.

For $y=\frac{1}{x}$, the vertical asymptote is $x=0$.

Behaviour at Infinity

As $x\to \pm\infty$, $\frac{1}{x}\to 0$. The graph approaches, but does not reach, the horizontal line $y=0$.

Note

Definition: Horizontal Asymptote

A horizontal asymptote is a horizontal line $y=c$ that a graph approaches as $x\to +\infty$ and/or $x\to -\infty$.

For $y=\frac{1}{x}$, the horizontal asymptote is $y=0$.

Relationship Between Domain, Range, and Asymptotes

For $y=\frac{1}{x}$:

Domain: $x\in \mathbb{R},\ x\neq 0$
Range: $y\in \mathbb{R},\ y\neq 0$

Tip

For the basic reciprocal shape, "excluded $x$-value" corresponds to a vertical asymptote, and "excluded $y$-value" corresponds to a horizontal asymptote.

The Transformed Reciprocal Function

A very important rational family is

$$f(x)=\frac{a}{x-h}+k \quad (a\neq 0).$$

You can think of this as transforming the parent function $\frac{1}{x}$ using:

translations (shifts) controlled by $h$ and $k$
a dilation (stretch/compression) and possible reflection controlled by $a$

Identifying Asymptotes

For

$$f(x)=\frac{a}{x-h}+k,$$

the vertical asymptote is $x=h$
the horizontal asymptote is $y=k$

This is one of the main reasons this form is so useful.

Identifying Domain and Range

Because $x\neq h$ (the denominator cannot be 0):

Domain: $x\in \mathbb{R},\ x\neq h$

Because the graph approaches but does not reach $y=k$:

Range: $y\in \mathbb{R},\ y\neq k$

Common Mistake

Students often think the curve "touches" the asymptote somewhere far away. It never does. Approaching an asymptote means getting arbitrarily close, not reaching it.

Effects of Parameters

Horizontal Translation

In $\frac{a}{x-h}+k$, the $(x-h)$ causes a horizontal translation:

if $h>0$, the graph shifts right by $h$
if $h<0$, the graph shifts left by $|h|$

The vertical asymptote moves from $x=0$ to $x=h$.

Vertical Translation

The $+k$ causes a vertical translation:

if $k>0$, the graph shifts up by $k$
if $k<0$, the graph shifts down by $|k|$

The horizontal asymptote moves from $y=0$ to $y=k$.

Vertical Stretch and Reflection

The parameter $a$ affects the "steepness" and orientation:

$|a|>1$ gives a vertical stretch (branches sit farther from the horizontal asymptote for the same $x$-distance)
$0<|a|<1$ gives a vertical compression
$a<0$ reflects the graph (relative to its asymptotes), swapping which quadrants the branches occupy around the centre point $(h,k)$

Analogy

Think of $(h,k)$ as moving the "crosshair" formed by the asymptotes, and $a$ as controlling how tightly the curve hugs that crosshair and which side it sits on.

Sketching Method

When you need a quick, accurate sketch of $y=\frac{a}{x-h}+k$, use a consistent routine.

Find asymptotes: draw dashed lines $x=h$ and $y=k$.
Decide orientation using the sign of $a$:
- if $a>0$, the branches lie in the "same orientation" as $\frac{1}{x}$ relative to the asymptotes
- if $a<0$, they are "flipped"
Plot one point on each side of the vertical asymptote by substituting convenient $x$-values (such as intercepts).
Draw smooth branches approaching both asymptotes.

Example

Sketch $y=\frac{1}{x-1}+3$.

- Vertical asymptote: $x=1$

- Horizontal asymptote: $y=3$

Test one value left of $x=1$: $x=0\Rightarrow y=\frac{1}{-1}+3=2$.

Test one value right of $x=1$: $x=2\Rightarrow y=\frac{1}{1}+3=4$.

Sketch the two branches passing near $(0,2)$ and $(2,4)$, each approaching the asymptotes.

Exam technique

In an exam sketch, the asymptotes should be clearly shown (usually dashed) and labeled. Then use at least two well-chosen points to anchor the curve. This earns method marks even if the final drawing is not perfect.

Inverses of Rational Functions

Note

Definition: Inverse Function

The inverse function of $f$, written $f^{-1}$, is the function that reverses $f$: if $f(a)=b$, then $f^{-1}(b)=a$. Graphically, $f$ and $f^{-1}$ are reflections across the line $y=x$.

Inverse of the Basic Function

Let $y=\frac{a}{x}$ with $a\neq 0$. Swap $x$ and $y$:

$$x=\frac{a}{y}.$$

Solve for $y$:

$$y=\frac{a}{x}.$$

$$f^{-1}(x)=\frac{a}{x}.$$

This means $\frac{a}{x}$ is an example of a function that is its own inverse.

Also note:

domain of $f^{-1}$: $x\neq 0$
range of $f^{-1}$: $y\neq 0$

Note

The reciprocal function $y=\frac{1}{x}$ is its own inverse, but it is not "the inverse of $y=x$." The inverse of $y=x$ is $y=x$ again.

Inverse of the Transformed Function

You can find the inverse by swapping $x$ and $y$ and solving for $y$. This is a common template:

Start: $$y=\frac{a}{x-h}+k$$

Swap: $$x=\frac{a}{y-h}+k$$

Then rearrange to isolate $y$.

Example

Find the inverse of $f(x)=\frac{-2}{x+8}-10$.

$y=\frac{-2}{x+8}-10$

Swap $x$ and $y$:

$x=\frac{-2}{y+8}-10$

Solve:

$x+10=\frac{-2}{y+8}$

$y+8=\frac{-2}{x+10}$

$y=\frac{-2}{x+10}-8$

So $$f^{-1}(x)=\frac{-2}{x+10}-8.$$

Common Mistake

After finding an inverse, students sometimes forget domain restrictions. If the inverse has a denominator, it also cannot be 0.

Applications and Significance

Reciprocal-type relationships appear whenever a quantity depends on "one over" another quantity, for example in scaling laws and engineering constraints. In structural engineering, understanding how values grow rapidly near a restricted input (a vertical asymptote) can help engineers anticipate extreme forces or design limits.

Theory of Knowledge

Graphs of rational functions show that mathematics can model "infinite behaviour" in a finite picture, by using asymptotes. To what extent is infinity in mathematics a real feature of the world, versus a useful concept that helps humans describe patterns and make predictions?

Self-Review Checklist

Active recall

1. Can you state the domain and range of $y=\frac{1}{x}$ and explain why 0 is excluded?

2. For $y=\frac{a}{x-h}+k$, can you write the asymptotes immediately and use them to state domain and range?

3. Can you explain what changing $a$, $h$, and $k$ does to the graph?

4. Can you find an inverse by swapping $x$ and $y$ and solving, and then check restrictions?

Rational Functions and Restrictions

Note

Definition: Rational Function

A rational function is any function that can be written as a quotient $\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x)\neq 0$.

A key "parent" example is the reciprocal function:

Note

Definition: Reciprocal Function

The reciprocal function is $f(x)=\frac{1}{x}$ with $x\neq 0$.

Its domain excludes 0 because division by 0 is undefined.

Note

Properties of the Reciprocal Function

The graph of $y=\frac{1}{x}$ has two separate curves (often called branches): one in Quadrant I and one in Quadrant III.

Behaviour Near Zero

As $x$ gets closer to 0:

as $x\to 0^+$, $\frac{1}{x}\to +\infty$
as $x\to 0^-$, $\frac{1}{x}\to -\infty$

So the graph gets closer and closer to the vertical line $x=0$ but never touches it.

Note

Definition: Vertical Asymptote

A vertical asymptote is a vertical line $x=c$ that a graph approaches as the input approaches $c$. For rational functions, it often occurs where the denominator approaches 0.

For $y=\frac{1}{x}$, the vertical asymptote is $x=0$.

Behaviour at Infinity

As $x\to \pm\infty$, $\frac{1}{x}\to 0$. The graph approaches, but does not reach, the horizontal line $y=0$.

Note

Definition: Horizontal Asymptote

A horizontal asymptote is a horizontal line $y=c$ that a graph approaches as $x\to +\infty$ and/or $x\to -\infty$.

For $y=\frac{1}{x}$, the horizontal asymptote is $y=0$.

Relationship Between Domain, Range, and Asymptotes

For $y=\frac{1}{x}$:

Domain: $x\in \mathbb{R},\ x\neq 0$
Range: $y\in \mathbb{R},\ y\neq 0$

Tip

For the basic reciprocal shape, "excluded $x$-value" corresponds to a vertical asymptote, and "excluded $y$-value" corresponds to a horizontal asymptote.

The Transformed Reciprocal Function

A very important rational family is

$$f(x)=\frac{a}{x-h}+k \quad (a\neq 0).$$

You can think of this as transforming the parent function $\frac{1}{x}$ using:

translations (shifts) controlled by $h$ and $k$
a dilation (stretch/compression) and possible reflection controlled by $a$

Identifying Asymptotes

For

$$f(x)=\frac{a}{x-h}+k,$$

the vertical asymptote is $x=h$
the horizontal asymptote is $y=k$

This is one of the main reasons this form is so useful.

Identifying Domain and Range

Because $x\neq h$ (the denominator cannot be 0):

Domain: $x\in \mathbb{R},\ x\neq h$

Because the graph approaches but does not reach $y=k$:

Range: $y\in \mathbb{R},\ y\neq k$

Common Mistake

Students often think the curve "touches" the asymptote somewhere far away. It never does. Approaching an asymptote means getting arbitrarily close, not reaching it.

Effects of Parameters

Horizontal Translation

In $\frac{a}{x-h}+k$, the $(x-h)$ causes a horizontal translation:

if $h>0$, the graph shifts right by $h$
if $h<0$, the graph shifts left by $|h|$

The vertical asymptote moves from $x=0$ to $x=h$.

Vertical Translation

The $+k$ causes a vertical translation:

if $k>0$, the graph shifts up by $k$
if $k<0$, the graph shifts down by $|k|$

The horizontal asymptote moves from $y=0$ to $y=k$.

Vertical Stretch and Reflection

The parameter $a$ affects the "steepness" and orientation:

$|a|>1$ gives a vertical stretch (branches sit farther from the horizontal asymptote for the same $x$-distance)
$0<|a|<1$ gives a vertical compression
$a<0$ reflects the graph (relative to its asymptotes), swapping which quadrants the branches occupy around the centre point $(h,k)$

Analogy

Think of $(h,k)$ as moving the "crosshair" formed by the asymptotes, and $a$ as controlling how tightly the curve hugs that crosshair and which side it sits on.

Sketching Method

When you need a quick, accurate sketch of $y=\frac{a}{x-h}+k$, use a consistent routine.

Find asymptotes: draw dashed lines $x=h$ and $y=k$.
Decide orientation using the sign of $a$:
- if $a>0$, the branches lie in the "same orientation" as $\frac{1}{x}$ relative to the asymptotes
- if $a<0$, they are "flipped"
Plot one point on each side of the vertical asymptote by substituting convenient $x$-values (such as intercepts).
Draw smooth branches approaching both asymptotes.

Example

Sketch $y=\frac{1}{x-1}+3$.

- Vertical asymptote: $x=1$

- Horizontal asymptote: $y=3$

Test one value left of $x=1$: $x=0\Rightarrow y=\frac{1}{-1}+3=2$.

Test one value right of $x=1$: $x=2\Rightarrow y=\frac{1}{1}+3=4$.

Sketch the two branches passing near $(0,2)$ and $(2,4)$, each approaching the asymptotes.

Exam technique

Inverses of Rational Functions

Note

Definition: Inverse Function

The inverse function of $f$, written $f^{-1}$, is the function that reverses $f$: if $f(a)=b$, then $f^{-1}(b)=a$. Graphically, $f$ and $f^{-1}$ are reflections across the line $y=x$.

Inverse of the Basic Function

Let $y=\frac{a}{x}$ with $a\neq 0$. Swap $x$ and $y$:

$$x=\frac{a}{y}.$$

Solve for $y$:

$$y=\frac{a}{x}.$$

$$f^{-1}(x)=\frac{a}{x}.$$

This means $\frac{a}{x}$ is an example of a function that is its own inverse.

Also note:

domain of $f^{-1}$: $x\neq 0$
range of $f^{-1}$: $y\neq 0$

Note

The reciprocal function $y=\frac{1}{x}$ is its own inverse, but it is not "the inverse of $y=x$." The inverse of $y=x$ is $y=x$ again.

Inverse of the Transformed Function

You can find the inverse by swapping $x$ and $y$ and solving for $y$. This is a common template:

Start: $$y=\frac{a}{x-h}+k$$

Swap: $$x=\frac{a}{y-h}+k$$

Then rearrange to isolate $y$.

Example

Find the inverse of $f(x)=\frac{-2}{x+8}-10$.

$y=\frac{-2}{x+8}-10$

Swap $x$ and $y$:

$x=\frac{-2}{y+8}-10$

Solve:

$x+10=\frac{-2}{y+8}$

$y+8=\frac{-2}{x+10}$

$y=\frac{-2}{x+10}-8$

So $$f^{-1}(x)=\frac{-2}{x+10}-8.$$

Common Mistake

After finding an inverse, students sometimes forget domain restrictions. If the inverse has a denominator, it also cannot be 0.

Applications and Significance

Theory of Knowledge

Self-Review Checklist

Active recall

1. Can you state the domain and range of $y=\frac{1}{x}$ and explain why 0 is excluded?

2. For $y=\frac{a}{x-h}+k$, can you write the asymptotes immediately and use them to state domain and range?

3. Can you explain what changing $a$, $h$, and $k$ does to the graph?

4. Can you find an inverse by swapping $x$ and $y$ and solving, and then check restrictions?

MYP Extended Mathematics Rational Functions Notes

MYP Extended Mathematics Rational Functions Notes

Rational Functions and Restrictions

Properties of the Reciprocal Function

Behaviour Near Zero

Behaviour at Infinity

Relationship Between Domain, Range, and Asymptotes

The Transformed Reciprocal Function

Identifying Asymptotes

Identifying Domain and Range

Effects of Parameters

Horizontal Translation

Vertical Translation

Vertical Stretch and Reflection

Sketching Method

Inverses of Rational Functions

Inverse of the Basic Function

Inverse of the Transformed Function

Applications and Significance

Self-Review Checklist

Rational Functions and Restrictions

Properties of the Reciprocal Function

Behaviour Near Zero

Behaviour at Infinity

Relationship Between Domain, Range, and Asymptotes

The Transformed Reciprocal Function

Identifying Asymptotes

Identifying Domain and Range

Effects of Parameters

Horizontal Translation

Vertical Translation

Vertical Stretch and Reflection

Sketching Method

Inverses of Rational Functions

Inverse of the Basic Function

Inverse of the Transformed Function

Applications and Significance

Self-Review Checklist

MYP Extended Mathematics Rational Functions Notes

Advanced Number and Bounds

Inequalities and Modelling

Function Transformations

Advanced Probability

Geometric Transformations

Variation and Standard Deviation

Trigonometric Functions and Rules

Sequences, Rational Expressions and Equations

Covariance and Correlation

Logarithms and Non-Linear Inequalities

Graph Theory and Vectors

MYP Extended Mathematics Rational Functions Notes

Advanced Number and Bounds

Inequalities and Modelling

Function Transformations

Advanced Probability

Geometric Transformations

Variation and Standard Deviation

Trigonometric Functions and Rules

Sequences, Rational Expressions and Equations

Covariance and Correlation

Logarithms and Non-Linear Inequalities

Graph Theory and Vectors

Rational Functions and Restrictions

Properties of the Reciprocal Function

Behaviour Near Zero

Behaviour at Infinity

Relationship Between Domain, Range, and Asymptotes

The Transformed Reciprocal Function

Identifying Asymptotes

Identifying Domain and Range

Effects of Parameters

Horizontal Translation

Vertical Translation

Vertical Stretch and Reflection

Sketching Method

Inverses of Rational Functions

Inverse of the Basic Function

Inverse of the Transformed Function

Applications and Significance

Self-Review Checklist

Rational Functions and Restrictions

Properties of the Reciprocal Function

Behaviour Near Zero

Behaviour at Infinity

Relationship Between Domain, Range, and Asymptotes

The Transformed Reciprocal Function

Identifying Asymptotes

Identifying Domain and Range

Effects of Parameters

Horizontal Translation

Vertical Translation

Vertical Stretch and Reflection

Sketching Method

Inverses of Rational Functions

Inverse of the Basic Function

Inverse of the Transformed Function

Applications and Significance

Self-Review Checklist