MATHEMATICS As LEVEL(FORM FIVE) NOTES – FUNCTION AND RELATION

FUNCTION AND RELATION

FUNCTIONS:

Is the correspondence between two objects. E.g., older than, smaller than, etc.

Relation can be thought of as:

• Rule
• A mapping
• A graph on the x-y plane

Example:

DOMAIN

Is the set of all possible values of x for which the corresponding value of y is known.

Example:

Given y =

RANGE

Is the set of all possible values of y for which the corresponding value of x is known.

Example:

Y = 2x

Relation as a mapping

In x-y plane (ordered pair):

FUNCTION:

Is the mapping of a single element from domain into range?

Not function

TYPES OF FUNCTION

The following are some types of functions:

1. CONSTANT FUNCTION
   f(x) = c
2. LINEAR FUNCTION
   f(x) = ax + b
3. QUADRATIC FUNCTION
   f(x) = ax² + bx + c
4. ABSOLUTE VALUE FUNCTION
   f(x) = |x|
5. RATIONAL FUNCTION
   

1: CONSTANT FUNCTION

SKETCHING THE FUNCTION:

Suppose given the function:

If f(x) = y

Given the function:

Suppose y = x for which x > 0

Solution:

Suppose that f(x) = y

y = x² – 1, x > 0

Step function

Sometimes referred to as compound functions, these are linear functions whose variables have a special relationship under certain conditions that make their graphs break into intervals (look like steps). To understand the concept, let us look at the following example.

The cost of shaving the hair of different age groups in a central salon is as follows:

• a) Shaving the group aged ten to twenty years costs Tsh 2000/=
• b) The group aged between twenty and thirty (exclusive) costs Tsh 4000/=
• c) The group aged thirty and above costs Tsh 6000/=

From the above information, provide domain and range.

Solution

If we let x be the ages and f(x) be the costs, then we can interpret this problem as a step function defined by:

The domain of this function is the set of real numbers such that x ≥ 10.

The range of this function is {2000, 4000, 6000}.

Graph of Quadratic function

A quadratic function is a polynomial of the second degree.

It is a function of the general form ax² + bx + c

Where a, b, and c are real numbers and a ≠ 0.

Example

Draw the graph of the function:

1. f(x) = x² – 1
2. f(x) = -x² – 1

Solution

Table Value

(i) f(x) = x² – 1

Its graph:

Its graph:

Drawing graph of cubic function

When the polynomial function is reduced to the third degree, a cubic function is obtained.

The cubic function takes the general form f(x) = ax³ + bx² + cx + d

Where a, b, c, and d are real numbers and a ≠ 0.

Example

Draw the graph of the following function:

f(x) = x³ – 9x

– The intercepts are points (-3,0), (0,0), (3,0)

– There are two turning points; the maximum at (-2,10) and the minimum at (2,-10)

– The domain is the set of all real numbers

– The range is the set of all real numbers y.

For the turning point, let us consider the function f(x) = ax² + bx + c.

The function f may be expressed in the form of g(a[g(x)] + k).

Where g(x) is another function in x and k is a constant as follows.

f(x) = ax² + bx + c

Factorizing out the constant a:

Example

Sketch the graph of f(x) = x² + 2x + 8, determine the turning point and the intercepts.

Solution

x² + 2x + 8 = 0

Solving we get:

(x + 2)(x – 4) = 0

x + 2 = 0, x – 4 = 0

x = -2, x = 4 which are intercepts

– The y-intercept C is 8

– To obtain the turning points, equate x² + 2x + 8 = 0 to ax² + bx + c = 0, so that the comparison we get:

a = 1, b = 2, c = 8

ASYMPTOTES

There are lines which the curve does not touch. There are two types of asymptotes:

1. Vertical asymptotes
2. Horizontal asymptotes

VERTICAL ASYMPTOTES (V.A)

Is the one which:

HORIZONTAL ASYMPTOTES

Is the one which:

RATIONAL FUNCTION SKETCH

Horizontal asymptote (H.A)

Sketch the function

Horizontal asymptotes.

Intercepts

Sketch.

Intercepts

Composite Function

Two functions f and g are said to be composite functions if fog = f(g(x)).

NOTE: COMMUTATIVE PROPERTY

Given f(x) = x² + 1 and g(x) = 2x.

Find (i) fog(x)

(ii) gof(x)

Approach: f(2x) = 2(x² + 1)

1. fog(x) = f(g(x)) = f(2x) = (2x)² + 1 = 4x² + 1

2. gof(x) = g(f(x)) = g(x² + 1) = 2(x² + 1)

CONCLUSION

fog ≠ gof, hence the composite function is not commutative.

ASSOCIATIVE PROPERTY

Given:

f(x) = x² – 1, g(x) = 3x, and h(x) = 2/x

(i) (fog) oh

(ii) fo (goh)

fog = f(g(x)) = f(3x) = (3x)² – 1 = 9x² – 1

Since fo(goh) = fo(goh), hence the composite function is associative.

FUNCTION

A function is a function when the line parallel to the y-axis cuts only once on the curve.

The line parallel to the x-axis cuts the curve only once.

An inverse function is one where each element from the domain matches exactly in the range, and conversely each element from the range matches exactly with the domain.

Given f(x) = 2x – 1

Find f^-1(x)

Approach

Sketch

(i) f(x) – state its Domain

(ii) f^-1(x)

Solution

f(x) = x + 1

Suppose f^-1(x) = g(x)

fog = f(g(x)) = x

gx + 1 = x

gx = x – 1