1. Functions: Domain and Range By Mr Porter
Function III -2 4
(1,-9)
axis 1
X-axis
Y-axis
y = x2 - 2x - 8
- 8
X-axis
Y-axis
y = mx + b b -b m
Functions: Domain and Range
By Mr Porter

2. Definitions Function: Domain Range
A function is a set of ordered pair in which no two ordered pairs have the same x-coordinate.
Domain
The domain of a function is the set of all x-coordinates of the ordered pairs.
[the values of x for which a vertical line will cut the curve.]
Range
The range of a function is the set of all y-coordinates 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. Linear Functions Any equation that can be written in the
General form ax + by + c = 0
Standard form y = mx + b
Sketching Linear Functions.
Find the x-intercep at y = 0
And the y-intercept at x = 0.
Examples
a) y = 3x + 6 b) 2x + 3y = 12
Y-axis
X-axis 4 6 -2 6
Y-axis
X-axis
y = 3x + 6
Every vertical line will cut 2x+3y =12.
x-intercept at y = 0
2x = 12
x = 6
y-intercept at x = 0
3y = 12
y = 4
x-intercept at y = 0
0 = 3x + 6
x = -2
y-intercept at x = 0
y = 6
Every vertical line will cut y = 3x + 6.
Every horizontal line will cut 2x+3y =12
Every horizontal line will cut y = 3x + 6

4. Special Lines Examples Vertical Lines: x = a
(4,5)
x = 4
X-axis
Y-axis
Vertical Lines: x = a
- these are not functions, as the first element in any ordered pair is (a, y)
a) x = 4
Equation of a vertical line is:
i) x = a
ii) x - a = 0
Domain: x = 4
(a,b)
x = a
X-axis
Y-axis
Range: all y in R
Sketch
b) x + 2 = 0
(-2,-6)
x = -2
X-axis
Y-axis
Domain: x = a
Range: all y in R
Domain: x = -2
Range: all y in R

5. Special Lines Examples Horizontal Lines: y = a
(-5,3)
y = 3
X-axis
Y-axis
Horizontal Lines: y = a
- these are functions, as the first element in any ordered pair is (x, a)
a) y = 3
Equation of a horizontal line is:
i) y = a
ii) y - a = 0
Domain: all x in R
(a,b)
y = a
X-axis
Y-axis
Range: y = 3
Sketch
b) y + 6 = 0
(2,-6)
y = -6
X-axis
Y-axis
Domain: all x in R
Range: y = b
Domain: all x in R
Range: y = -6

6. Parabola: y = ax2 +bx + c Example
Sketch y = x2 + 2x - 3, hence, state its domain and range.
The five steps in sketching a parabola function:
1) If a is positive, the parabola is concave up.
If a is negative, the parabola is concave down.
2) To find the y-intercept, put x = 0.
3) To find the x-intercept, form a quadratic and solve
ax2 + bx + c = 0
* factorise
* quadratic formula
4) Find the axis of symmetry by
5) Use the axis of symmetry x-value
to find the y-value of the vertex, h
1) For y = ax2 + bx + c
a = 1, b = +2, c = Concave-up a = 1
2) y-intercept at x = 0, y = -3
3) x-intercept at y = 0, (factorise )
(x - 1)(x + 3) = 0
x = +1 and x = - 3.
4) Axis of symmetry at = -1
5) y-value of vertex: y = (-1)2 +2(-1) - 3
y = -4 -3 1
(-1,-4)
X-axis
Y-axis -1
Domain: all x in R
Range: y ≥ -4
Domain: all x in R
Range: y ≥ h for a > 0
Range: y ≤ h for a < 0

7. Parabola: y = ax2 +bx + c Example
Sketch y = –x2 + 4x - 5, hence, state its domain and range.
The five steps in sketching a parabola function:
1) If a is positive, the parabola is concave up.
If a is negative, the parabola is concave down.
2) To find the y-intercept, put x = 0.
3) To find the x-intercept, form a quadratic and solve
ax2 + bx + c = 0
* factorise
* quadratic formula
4) Find the axis of symmetry by
5) Use the axis of symmetry x-value
to find the y-value of the vertex, h
1) For y = ax2 + bx + c, a = -1, b = +4,
c = -5. Concave-down a = -1
2) y-intercept at x = 0, y = -5
3) x-intercept at y = 0, NO zeros by
Quadratic formula.
4) Axis of symmetry at = +2
5) y-value of vertex: y = -(2)2 +4(2) - 5
y = -1
(2,-1) -5
X-axis
Y-axis 2
Domain: all x in R
Range: y ≤ -1
Domain: all x in R
Range: y ≥ h for a > 0
Range: y ≤ h for a < 0

8. Step 1: Determine concavity: Up or Down?
Worked Example 1: Your task is to plot the key features of the given parabola, sketch the parabola, then state clearly its domain and range.
Sketch the parabola y = x2 - 2x - 8, hence state clearly its domain and range. -2 4
(1,-9)
axis 1
X-axis
Y-axis
Step 1: Determine concavity: Up or Down?
The five steps in sketching a parabola function:
1) If a is positive, the parabola is concave up.
If a is negative, the parabola is concave down.
2) To find the y-intercept,
put x = 0.
3) To find the x-intercept, form a quadratic and solve
ax2 + bx + c = 0
* factorise
* quadratic formula
Find the axis of symmetry by
5) Use the axis of symmetry x-value to find the y-value of the vertex, h
For the parabola of the form y = ax2 + bx + c
a = 1 => concave up
y = x2 - 2x - 8
Step 2: Determine y-intercept.
Let x = 0, y = -8
Step 3: Determine x-intercept.
Solve: x2 - 2x - 8 = 0
Factorise : (x - 4)(x + 2) = 0 ==> x = 4 or x = -2.
Step 4: Determine axis of symmetry.
- 8
Step 5: Determine maximum or minimum y-value (vertex).
Substitute the value x = 1 into y = x2 - 2x - 8.
Domain all x in R
Range y ≥ -9
y = (1)2 - 2(1) - 8 = -9
Vertex at (1, -9)

9. Step 1: Determine concavity: Up or Down?
Worked Example 2: Your task is to plot the key features of the given parabola, sketch  the parabola, then state clearly its domain and range.
Sketch the parabola f(x) = x - x2, hence state clearly its domain and range. -5 3
(1,16)
axis -1
X-axis
Y-axis
Step 1: Determine concavity: Up or Down?
The five steps in sketching a parabola function:
1) If a is positive, the parabola is concave up.
If a is negative, the parabola is concave down.
2) To find the y-intercept,
put x = 0.
3) To find the x-intercept, form a quadratic and solve
ax2 + bx + c = 0
* factorise
* quadratic formula
Find the axis of symmetry by
5) Use the axis of symmetry x-value to find the y-value of the vertex, h
For the parabola of the form f(x) = ax2 + bx + c
a = -1 => concave down
Step 2: Determine y-intercept.
f(x)=15 - 2x - x2 15
Let x = 0, f(x) = +15
Step 3: Determine x-intercept.
Solve: x - x2 = 0
Factorise : (3 - x)(x + 5) = 0 ==> x = 3 or x = -5.
Step 4: Determine axis of symmetry.
Step 5: Determine maximum or minimum y-value (vertex).
Substitute the value x = -1 into y = x - x2.
Domain: all x in R
Range: y ≤ 16
y = (-1) - (-1)2 = 16
Vertex at (1, 16)

10. Exercise: For each of the following functions: a) sketch the curve
b) sate the largest possible domain and range of the function.
(i) f(x) = 5 - 2x
(ii) h(x) = 2x2 + 7x - 15
Domain: All x in R
Range: All y in R
Domain: All x in R
Range: All y ≥ -211/8 -5
11/2
-13/4
Y-axis
X-axis
h(x) = 2x2 + 7x - 15
-15
(-13/4 ,-211/8 )
X-axis
Y-axis
f(x) = 5 - 2x 5
21/2
(iii) h(x) = x2 + 2x + 5
(iv) g(x) = 5x + 4
NO x-intercepts.
(try quadratic formula?) -1
Y-axis
X-axis
h(x) = x2 + 2x + 5 5
(-1 ,4)
Domain: All x in R
Range: All y ≥ 4
Domain: All x in R
Range: All y in R
X-axis
Y-axis
g(x) = 5x + 4 4
-4/5