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.
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:
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.
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.
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 $y=0$.
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: $y\in \mathbb{R},\ y\neq 0$
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: $y\in \mathbb{R},\ y\neq k$
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 $(h,k)$
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.
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.
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
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 $y$:
$$y=\frac{a}{x}.$$
So
$$f^{-1}(x)=\frac{a}{x}.$$
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 $f^{-1}$: $y\neq 0$
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 $y$.
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 $$f^{-1}(x)=\frac{-2}{x+10}-8.$$
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.
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
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?
