Chapter 3: Functions

Question 1

What is the difference between relations and functions?
How can we test for functions?

Answer 1

• Relation: Any relationship between the x and y variables (usually in the form of an equation that connects the variables through points)
• Function: Relation where no 2 points have the same y-coordinate (1 x-coordinate cannot have multiple y-coordinates)

Test for functions:
1) Algebraic test: sub any x-value
- 1 y-value: Function
- Multiple y-values: Relation

2) Vertical line test: draw all the possible vertical lines on the graph
- Every line only cuts graph once: Function
- Line cuts graph more than once: Relation

Question 2

Define domain and range.
State the different ways domain and range can be notated

Answer 2

Domain: Range of possible x-values
Range: Range of possible y-values

Notation:
1) Set notation
Eg. {xl -2<x≤4}
2) Interval notation
Eg. 2<x≤4
3) Number line notation
(Open-circle: Point not included)
(Filled circle: Point included)
(Arrowhead: Continues indefinitely)

Question 3

State the natural domains and ranges of functions
(x², √x, 1/x, 1/√x)

Answer 3

1) x²
Natural domain: x∈R

2) √x
Natural domain: x≥0

3) 1/x
Natural domain: x≠0

4) 1/√x
Natural domain: x>0

Question 4

Define a rational function

Answer 4

Rational function: when a polynomial is divided by another polynomial

Question 5

Draw the graph for and state the asymptotes of a reciprocal function
(y=1/x)

Answer 5

Graph for *:
- y=1/x
- y= -1/x

Horizontal asymptote: y=0
Vertical asymptote: x=0

Question 6

Find the horizontal and vertical asymptotes of rational functions in forms:
y=(b/cx+d) + a
y= ax+b/cx+d

Answer 6

y=(b/cx+d) + a:
- Horizontal asymptote: a
- Vertical asymptote: -d/c

y= ax+b/cx+d
- Horizontal asymptote: a/c
- Vertical asymptote: -d/c

Question 7

Solve composite functions*

Answer 7

f◦g = f (g(x) )
- x of f(x) = g(x)

g◦f = g (f(x) )
- x of g(x) = f(x)

f◦f = f (f(x) )
- x of f(x) = f(x)

g◦g = g (g(x) )
- x of g(x) = g(x)

Question 8

Find inverse functions* and state the properties of an inverse graph

Answer 8

Inverse function: f⁻¹(x)
- Let y=f(x)
- Make x the subject of the equation
- Replace y with x to find f⁻¹(x)

Properties of inverse function graph:
1) f⁻¹(x) is the reflection of f(x) at y=x
(x,y) –> (y,x)

2) Domain and range are inverted
- Domain of f(x) is the range of f⁻¹(x)
- Range of f(x) is the domain of f⁻¹(x)

3) (f◦f⁻¹)(x)=x