The Reciprocal Function
Note 1: The Reciprocal Function
The reciprocal of x is 1 or x-1
x
and x . x-1 = 1
k
The reciprocal of x is f(x) =
where k is a
x
constant.
Graphs of reciprocal functions have similar shapes.
1
f(x) =
x
Note 2: Asymptotes
Is the straight line that a curve gets
continually closer to but never meets it.
y = f(x)
y=b
y = b is an asymptote of function f(x) and
reads:
As x
∞, f(x) b
The symbol
means ‘approaches’
The graph of a reciprocal function is called an
hyperbola
y = -x
y=x
y=





k
x
x-axis is the horizontal asymptote
y-axis is the vertical asymptote
Domain and range are all real numbers except 0
y = -x and y = x are the lines of symmetry for this
function
The reciprocal function is a self-inverse function
Example:
For each function :
Write the equations of the vertical and horizontal
asymptotes
Sketch graph
State domain and range
y=
9
x
Asymptotes: x = 0, y = 0
Domain: x and y ε R,
x and y ≠ 0
y=
9
+2
x
x = 0, y = 2
x and y ε R,
x ≠ 0, y ≠ 2
note: graph of f(x) + 2 is the
same as f(x) but shifted up 2 units
Note 3: Rational Functions
A rational function is in the form
g(x)
f (x) =
h(x)
where g and h are polynomials and h(x) ≠ 0, ie:
f (x) =
ax + b
cx + d
Investigation –Graphing
Rational Functions
Use your calculator to show sketches of
y=
1
x
y=
1
x-2
y=
1
x+3
Copy and complete the table:
Rational
Function
Vertical
Asymptote
Horizontal
Asymptote
Domain
Range
y = 1/x
y = 1/(x-2)
y = 1/(x+3)
What effect does changing the denominator have on the
vertical asymptote
What do you notice about:
•
the horizontal asymptote
• The domain and the value of the vertical asymptote
• The range and the value of the horizontal asymptote
worksheet
Note 4: Rational Functions in the form
y=
k
x-b
where k and b are constants


have a vertical asymptote when denominator = 0,
ie. when x = b
A horizontal asymptote is the x-axis, ie y = 0
Example:
Identify the vertical & horizontal asymptotes of
State the domain and range
Sketch the function
Vertical asymptote when x = 3
Horizontal asymptote when y = 0 (x-axis)
Domain x ε R, x ≠ 3
Range y ε R, y ≠ 0
y=
1
x -3
Note 5: Rational Functions in the form
y=


ax + b
cx + d
vertical asymptote occurs at the x value that
makes the denominator 0
The horizontal asymptote is the line
y=
a
c
Example:
x +1
For the function y = 2 x - 4
a)Sketch the graph
b)Find the vertical & horizontal asymptotes
c)State the domain and range
Vertical asymptote when 2x – 4 = 0
x=2
a
Horizontal asymptote when y = , so
c
Domain
x ε R, x ≠ 2
Range
y ε R, y ≠ ½
y=
1
2
worksheet