1. Function - Domain & Rangeh k
(h,k)
X-axis
Y-axis
f(x)=(x-3)2(x+4) r
– r
Cubic, Circle, Hyperbola, etc
By Mr Porter

2. Domain - independent variable
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.

3. 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

4. Graphs of Odd Power Functions : n = 3 or 5
X-axis
Y-axis
f(x)=-x(x-3)(x+4)
X-axis
Y-axis
f(x)=(x-3)2(x+4)
X-axis
Y-axis
f(x)=(x-3)(x+4)(x+1)(x-1)
Every vertical line and horizontal line will cut the curve.
X-axis
Y-axis
f(x)=(x+3)2(x-4)3
X-axis
Y-axis
f(x)=2x3
Hence,
Domain : all x in R
Range : all y in R

5. Circle There are to forms of the circle:
a) Standard circle centred at the origin (0,0) radius r.
x2 + y2 = r2
Domain: -r ≤ x ≤ r and Range: -r ≤ y ≤ r
b) General circle, centred at (h,k) with radius r.
(x - h)2 + (y - k)2 = r2
Domain: -r +h ≤ x ≤ r + h 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
2) x2 + y2 = 25
==> x2 + y2 = 52
Circle centred (0,0), radius r = 3 units
Circle centred (0,0), radius r = 5 units 3 -3
X-axis
Y-axis 5 -5
X-axis
Y-axis
Domain: -3 ≤ x ≤ 3
Range: -3 ≤ y ≤ 3
Domain: -5 ≤ x ≤ 5
Range: -5 ≤ y ≤ 5

6. Hyperbola As with the circle, there are two forms of the hyperbola:
a) Standard:
b) General:
centred (h,k) h k
(h,k)
Domain:
All x in R, x ≠ 0
Range:
All y in R, y ≠ 0
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.
These lines are called ASYMPTOTE lines.
Example: Find the asymptotes for
Hence, sketch.
Y-axis
X-axis
The vertical asymptote is found by setting the DENOMINATOR to zero and solving for x.
Vertical Asymptote.
Denominator : x - 3 = 0 ==> x = 3.
Horizontal Asymptote.
Let x = , y is almost 0, i.e. y = 0 3
Asymptotes -2 3
The horizontal asymptote is a little harder to find, at this stage, use a very large value of x, say x = , then round off for a good common sense estimate.
Domain: all x in R, x ≠ 3
Range: all y in R, y ≠ 0
The correct method is to use limits for the horizontal asymptote.

7. 1) Find the asymptotes for Hence, sketch. 2) Find the asymptotes for
Examples.
1) Find the asymptotes for
Hence, sketch.
2) Find the asymptotes for
Hence, sketch.
Vertical Asymptote: Denominator : x - 3 = 0 ==> x = 3.
Vertical Asymptote: Denominator : x - 3 = 0 ==> x = -2.
Horizontal Asymptote: Let x = , y is almost -0, i.e. y = 0
Horizontal Asymptote: Let x = , y is almost -0, i.e. y = 0
Also,
for , a < 0, 2nd & 4th Quadrants
Also,
for , a > 0, 1st & 3rd Quadrants
X-axis
Y-axis
Y-axis
X-axis 3 -2 2 3 2
Domain: all x in R, x ≠ 3
Range: all y in R, y ≠ 0
Domain: all x in R, x ≠ -2
Range: all y in R, y ≠ 0

8. Semi-Circles. The general form of a standard semi-circle is
Represents the top half
of the circle
Represents the bottom half
of the circle 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.

9. Exercise. For each of the following, sketch the function (curve), then clear write down the DOMAIN and RANGE.
X-axis
Y-axis
X-axis
Y-axis -4 4
Hint: Determine if the function is a:
Circle
Hyperbola
Semi-circle.
Hint: Determine if the function is a:
Circle
Hyperbola
Semi-circle.
Domain: all x in R, x ≠ 0
Range: all y in R, y ≠ 0
Domain: -4 ≤ x ≤ 4
Range: 0 ≤ y ≤ 4
X-axis
Y-axis 4 1 2
X-axis
Y-axis
2√3
-2√3
Hint: Determine if the function is a:
Circle
Hyperbola
Semi-circle.
Hint: Determine if the function is a:
Circle
Hyperbola
Semi-circle.
Domain: -2√3 ≤ x ≤ -2√3
Range: -2√3 ≤ y ≤ -2√3
Domain: all x in R, x ≠ 4
Range: all y in R, y ≠ 0