Function - Domain & Range
Y-axis
r
k
(h,k)
X-axis
0
0
h
f(x)=(x-3)2(x+4)
Cubic, Circle, Hyperbola, etc
By Mr Porter
–r
r
Definitions
Function:
A function is a set of ordered pair in which no two ordered pairs have the same
x-coordinate. E.g. (3,5) (2,-2) (4,5)
Domain - independent variable
The domain of a function is the set of all x-coordinates or first element of the
ordered pairs.
[the values of x for which a vertical line will cut the curve.]
Range - dependent variable
The range of a function is the set of all y-coordinates or the second element of the
ordered pairs.
[the values of y for which a horizontal line will cut the curve]
Note: Students need to be able to define the domain and range from the equation of a curve or function. It is encourage
that student make sketches of each function, labeling each key feature.
Cubic and Odd Power Functions.
A general function can be written as
f(x) = axn + bxn-1 + cxn-2 + …… + z
where a ≠ 0 and n is a positive integer.
If the power n is an ODD number, n = 3, 5, 7, …..
then f(x) is an odd powered function. We can generalise the
domain and range as follows:
Domain : All x in the real numbers, R.
Range : All y in the real numbers, R.
This is very true for the following functions:
f(x) = 2x3 + 4
g(x) = 5 - x5
h(x) = x3 - x2 + 5x -7
Graphs of Odd Power Functions : n = 3 or 5
Y-axis
Y-axis
X-axis
f(x)=-x(x-3)(x+4)
Y-axis
X-axis
X-axis
f(x)=(x-3)2(x+4)
f(x)=(x-3)(x+4)(x+1)(x1)
Y-axis
f(x)=(x+3)2(x-4)3
Every vertical line and
horizontal line will cut
the curve.
Y-axis
X-axis
X-axis
f(x)=2x3
Hence,
Domain : all x in R
Range : all y in R
Circle
There are to forms of the circle:
a) Standard circle centred at the origin (0,0) radius r.
Domain: -r ≤ x ≤ r
and
Range: -r ≤ y ≤ r
(x - h)2 + (y - k)2 = r2
b) General circle, centred at (h,k) with radius r.
Domain: -r +h ≤ x ≤ r + h
x2 + y2 = r2
and
Range: -r + k ≤ y ≤ r + k
All circles are RELATIONS, but by restricting the RANGE, we convert them to functions.
Standard Circle
1) x2 + y2 = 9 ==> x2 + y2 = 32
Circle centred (0,0), radius r = 3 units
2) x2 + y2 = 25 ==> x2 + y2 = 52
Circle centred (0,0), radius r = 5 units
Y-axis
Y-axis
3
5
3
X-axis
-3
-3
5
X-axis
-5
Domain: -3 ≤ x ≤ 3
Range: -3 ≤ y ≤ 3
-5
Domain: -5 ≤ x ≤ 5
Range: -5 ≤ y ≤ 5
Hyperbola
As with the circle, there are two forms of the hyperbola:
a) Standard:
a
y
x
Domain:
All x in R, x ≠ 0
Range:
All y in R, y ≠ 0

0
0
b) General: (y  k) 

k
(h,k)
0
0
h
a
(x  h)
centred (h,k)
Domain:
All x in R, x≠h
Range:
All y in R, y ≠ k
The curve does not cut a VERTICAL or a HORIZONTAL line.
2
These lines are called ASYMPTOTE lines.
Example: Find the asymptotes for y 
x3
The horizontal
Hence, sketch.
Y-axisasymptote is found by
setting the DENOMINATOR to zero and
Vertical Asymptote.
solving for x.
Denominator : x - 3 = 0 ==> x = 3.
The vertical asymptote is a little harder to
Horizontal Asymptote. 
0
find, at this stage, use3 a very large value
of
X-axis
-2
x, say x = 1 000
000, then round off for a
3
Asymptotes
good common sense estimate.
The correct method is to use limits for the horizontal asymptote.
Let x = 1 000 000, y is almost 0, i.e. y = 0
Domain: all x in R, x ≠ 3
Range: all y in R, y ≠ 0
Examples.
1) Find the asymptotes for
Hence, sketch.
y
2
x3
2) Find the asymptotes for
Hence, sketch.
y
4
x2
Vertical Asymptote: Denominator : x - 3 = 0 ==> x = 3.
Vertical Asymptote: Denominator : x - 3 = 0 ==> x = -2.
Horizontal Asymptote: Let x = 1 000 000, y is almost -0, i.e. y = 0
Horizontal Asymptote: Let x = 1 000 000, y is almost -0, i.e. y = 0

Also,
a
for y 
, a < 0, 2nd & 4th Quadrants
xh

Also,
a
for y 
, a > 0, 1st & 3rd Quadrants
xh

Y-axis
Y-axis
Domain: all x in R, x ≠ 3
Range: all y in R, y ≠ 0
2
2
3
X-axis
0
3
-2
0
X-axis
Domain: all x in R, x ≠ -2
Range: all y in R, y ≠ 0
Semi-Circles.
2
2
The general form of a standard semi-circle is y   r  x
y  r 2  x 2 Represents the top half
of the circle

y  r 2  x 2 Represents the bottom half
of the circle

r
–r
–r
r
r
–r
Domain: -r ≤ x ≤ r
Range: 0 ≤ y ≤ r
Domain: -r ≤ x ≤ r
Range: -r ≤ y ≤ 0
To sketch s semi-circle or circle, draw the semi-circle first, then label domain and range.
Exercise. For each of the following, sketch the function (curve), then clear write down
the DOMAIN and RANGE.
1) y 
5
x
Y-axis
Hint: Determine if the function
is a:
Circle
Hyperbola
Semi-circle.
X-axis
2) y  16  x
2
Y-axis
Hint: Determine if the function
4
is a:
Circle
Hyperbola
X-axis
Semi-circle.

Domain: all x in R, x ≠ 0
Range: all y in R, y ≠ 0
3) x  y  12
2
2
function
2√3
X-axis
2
4) y 
x4
Y-axis
Hint: Determine if the function
is a:
Circle 1
Hyperbola2
X-axis
Semi-circle.
4
2√3

-2√3
-2√3
Domain: -2√3 ≤ x ≤ -2√3
Range: -2√3 ≤ y ≤ -2√3
4
Domain: -4 ≤ x ≤ 4
Range: 0 ≤ y ≤ 4
Y-axis
Determine
if the
Hint:
is a:
Circle
Hyperbola
Semi-circle.
-4
Domain: all x in R, x ≠ 4
Range: all y in R, y ≠ 0