Reciprocal Functions
The reciprocal function is defined as $f(x) = \frac{1}{x}$, where $x \neq 0$. This function has several unique properties.
Graph of the Reciprocal Function
The graph of $f(x) = \frac{1}{x}$ consists of two separate branches:
- A branch in the first quadrant (where both x and y are positive)
- A branch in the third quadrant (where both x and y are negative)
The x-axis and y-axis are asymptotes for this function, which means the graph gets infinitely close to these lines but never touches them.
Self-Inverse Nature
One of the most interesting properties of the reciprocal function is that it is self-inverse. This means that if you apply the function twice, you get back to where you started.
Mathematically, this can be expressed as:
$f(f(x)) = x$
Let's verify this: $$f(x) = \frac{1}{x}$$ $$f(f(x)) = f(\frac{1}{x}) = \frac{1}{\frac{1}{x}} = x$$
This property is visually represented by the symmetry of the graph about the line $y = x$.
Simple Rational Functions
A simple rational function is of the form $f(x) = \frac{ax + b}{cx + d}$, where $a$, $b$, $c$, and $d$ are constants and $c \neq 0$.
Graph Characteristics
The graphs of rational functions can have various shapes depending on the values of $a$, $b$, $c$, and $d$. However, they all share some common features:
- Vertical asymptote
- Horizontal asymptote
- x-intercept (if it exists)
- y-intercept (if it exists)
Asymptotes
Asymptotes are lines that a graph approaches but never quite reaches.
Vertical Asymptotes
A vertical asymptote occurs where the denominator of the rational function equals zero.
For $f(x) = \frac{ax + b}{cx + d}$, the vertical asymptote is at $x = -\frac{d}{c}$.
For $f(x) = \frac{2x + 1}{x - 3}$, the vertical asymptote is at $x = 3$.
Horizontal Asymptotes
The horizontal asymptote shows the long-term behavior of the function as $x$ approaches infinity.
For $f(x) = \frac{ax + b}{cx + d}$, the horizontal asymptote is $y = \frac{a}{c}$.
For $f(x) = \frac{2x + 1}{x - 3}$, the horizontal asymptote is $y = 2$.
If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.
Sketching Rational Functions
When sketching rational functions, follow these steps:
- Find the vertical asymptote(s)
- Find the horizontal asymptote
- Find the x-intercept(s) (if any)
- Find the y-intercept
- Plot a few additional points
- Sketch the curve
Always check which side of the vertical asymptote the function approaches positive or negative infinity. This helps in accurately sketching the function.
Let's sketch $f(x) = \frac{x + 2}{x - 1}$
- Vertical asymptote: $x = 1$
- Horizontal asymptote: $y = 1$
- x-intercept: When $y = 0$, $x = -2$
- y-intercept: When $x = 0$, $y = -2$
- Additional points: $(2, \frac{4}{1} = 4)$, $(-1, \frac{1}{-2} = -\frac{1}{2})$
- Sketch the curve
Connection to Transformations
The function $f(x)\frac{1}{x}$ is a parent function, and it is necessary to learn how they behave under transformation.
$f(x) = \frac{2x + 1}{x - 3}$ can be seen as a transformation of $\frac{1}{x}$:
- Stretch vertically by a factor of 2
- Shift right by 3 units
- Shift up by $\frac{1}{3}$ units
Students often forget that the vertical asymptote moves in the opposite direction of the horizontal shift. In the example above, the -3 in the denominator moves the vertical asymptote 3 units to the right.
A form that allows you to see the asymptotes and transformations easily is
$$y=\frac{k}{a(x-c)}+h$$
Where $y=h$ is the horizontal asymptote and $x=c$ is the vertical asymptote, thus translating the parent graph $h$ units vertically and $c$ units horizontally.
