Relations and functionsA relation may be written as a set of ordered pairs. It shows a relationship between a two sets of items. Coordinates such as (2, 5) (4, 5) shows the relation between x values (ordinate) and y values (abscissa). The first entry is the domain and the second entry is the range.

Functions however, are special types of relations where there exist a one to one relation or a many to one relationship.

One to One

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Many to One

A one to many relation is not a function.

One to Many

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The above diagrams are known as arrow/mapping diagrams which shows the relationship between the domain and the range. One should be able to deduce an expression that maps the domain to the range.

Can you determine the expression for mapping the domain to the range?

Evaluating FunctionsFunctions may be evaluated by simply substituting the value of the domain within the expression for the function. For example given , Read equals three times x plus 3. Which means whatsoever, the domain value is, the range may be evaluated by substituting the value of x within the expression for . Hence

Therefore given a set of domain values one should be able to evaluate the corresponding values for the range. If the domain is , the values of x are, -2,-1, 0, 1, 2, and 3. The corresponding range values are -4, -1, 2, 5, 8 and 11. This may be written as a set of ordered pairs as (-2,-4) (-1, -1), (0, 2), (1,5) ,(2, 8) and (3, 11).

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Let us consider the function

Evaluate

Solution

Consider the flowing

and

Evaluate if if is undefined. if equals 0.

Solution

=

This requires us to calculate the value of x such that h(x) equals zero.

It therefore follows that

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For the function to be undefined the denominator equals zero. Therefore,

or

Exercise 1.Evaluate the following given the functions

, and

Evaluate if if if

Composite FunctionsComposite functions are the resulting of composing two or more simpler functions to form one expression.This may be accomplished by using the principles of substitution. Suppose that and .

Find an expression for

The diagram below may be used to model the composite functions.

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g(x) =

Likewise

Alternatively one may decide to use the principle of substituting one expression into the other, similarly to what was done for evaluating a function.

indicates that the expression for should be substitution. Hence == 2x +3

Likewise

Exercise 2.

Evaluate the following expressions using the functions below.

, and

Revision Questions

1. Given that f: x → 3x – 1 and g : x

(i) Evaluate f(3)

(ii) Evaluate g(4)

(iii) Derive an expression for fg(x)

(iv) Derive an expression for gf(x)

(iii) Hence calculate fg(10).

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4. Given that the cost of a basic pizza is $1000 and the cost of each additional

toppings is $150. Write an equation using p to represent the total cost of the

pizza and t to represent the information above.

Inverse of Functions

DOMAIN RANGE

The above diagram indicates the relationship between the domain and the range where domain values are mapped to the range by the expression 3x +1.The inverse of this function requires the range to be mapped to the domain, hence the process is reversed. Can you think of how the process will be reversed?

The reverse of this process requires all operations to be negated (doing the opposite and the order of the operations is in the opposite sequence).For the example above the reverse will require the subtraction of 1, then the result is to be divided by three. Proof (7 -1) ÷3 = 6÷3 = 2

SuggestionsThe easiest approach to finding the inverse requires one to be conscious of the different operations and their relationship, that is addition and subtraction are opposite to each other similarly, multiplication and division. The reverse process requires us to do the opposite of the last operation that mapped the domain to the range.

Lets Practice!

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This implies the multiplication operation (multiply the domain values by four). The opposite of multiplication is division. Hence, the inverse of h is to divide the values within the range by four to find the domain values.

This is written as

The reverse of this requires the addition of 1, then the division of the result by 4.

This is expressed as

. Recall that means x is squared first then multiplied by 2 and the result, 1 is added to it.Hence the reverse of this operation is to subtract 1, divide by two then take the square root.

This is written as

Exercise 3. Write an expression for the inverse of these functions

Graph of FunctionsThe graph of a function may be drawn by plotting and drawing the ordered pairs. Remember that all ordered pairs are not function, hence not all graphs are function. However, all graphs represent a relation.

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Determine which of the following represent a function clearly stating a reason(a)

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(b)

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Y= x̂ 2 +1

(c)

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Y= x̂ -1 -1

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-2y = x +3(e)

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y =x3-5Can you determine another way which would be easier to determine whether the graph is function or not?

Solution Exercise 3.

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Drawing a graph of a function

Given that and , plot on draw the graph of using a scale of 1cm to represent one unit on both axis.

It would be suitable to create a table of values which would be easier for you to read the ordered pairs, see table below.

X -2 -1 0 1 2 3Y 3

To complete the table one is required to evaluate the function as x varies.Hence, when x = -2,

when x = -1 when x = 1, when x = 2, when x = 3,

Marks will be given for completing the table of values and not necessarily for showing the calculations.

The graph is illustrated overleaf, please attend to some of the characteristics of the line.

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-2y =2x+3Y = 2x + 3

The graph of other functions may be created in the similar way that is by completing the table of values.

Example 2. Draw the graph of , for .

X -2 -1 0 1 2 3Y 6

x =-2, x = -1, x = 1, x = 2, x =3,

See graph below

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Worksheet

1. Given that, the functions f(x) and g(x) are defined such that

and

(i) Evaluate

(ii) Evaluate g(10)

(iii) Evaluate

(iv) Write an expression for fg(x)

(v) Find the value of x for which f(x) =10

(vi) Determine

2. Given that f: x → 3x – 1 and g : x

(v) Evaluate f(3)

(vi) Evaluate g(4)

(vii) Derive an expression for fg(x)

(viii) Derive an expression for gf(x)

(iii) Hence calculate fg(10)

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3. Given that f: x → 3 – x and

(i) Calculate g(2)

(ii) Derive an expression for g f (x)

(iii) Calculate the value of f -1 (4)

4. Evaluate the following functions for x = -2

(i) f(x) = 2x2 +5x +3

(ii) g(x) =-x2 +x -6

Section B

5. By completing the table of values below, plot the graph of for

the domain .

X 0 1 2 3 4 5 6

Y 6 -5 4

Use this graph to solve the following equation

6. Given that y =2x2 + x, copy and complete the table below

(i)

x -1 0 1 2 3 4

y

(ii) Draw the graph of the function y = 2x2 + x for -1 ≤ x ≤4.

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Worksheet 2College Math- Functions and Relations

Multiple Choice- Choose the correct response

1. If f(x) = x +2, then f(x2) is(a) x2 +2 (b) x2 +4 (c) x2 +4x +4 (d) 2x + 4

2. The mapping diagram below describes the relation

(a) x is greater than y (b) x is a multiple of y (c) x is divisible by y (d) x is a factor of y

3. If and then (a) -13 (b) -7 (c) 5 (d) 20

4. Which of the following represents the relation

(a)(0, 3), (1, 4), (2, 7), (3,12) (b) (0, 3) (1, 5), (2, 7), (3, 12)(c ) (0, 3), (1,4), (2, 5), (3, 6) (d) (0, 3), (1, 1), (2, 4), (3, 9)

5. Which one of the following ordered pairs describes a function?

(a) (2,3), (2, 5), (4, 7) (b)(2,1), (4, 3), (5, 7)( c) (3,-2) (2, 4), (3, 6) (d)(-1,4), (-1, 5), (2, 5)

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6. The arrow diagram below shows a function. Which of the following best describes the function?

(a) (b) (c) (d)

7. If , then (a) -5 (b) -8 (c) 7 (d) 10

8. If and , then

(a) 14 (b) 17 (c) 25 (d) 45

9. If , then the image of -2 is

(a) (b) (c) (d)

10. If , then is

(a) (x +5) (b) (c)

(d)

11. If then f(-2) is (a) 3 (b) 9 (c) -9 (d) 7

12. If , then is(a) 0 (b) 6 (c) -6 (d) -3

13. Given that and , then g(f(x)

(a) 6x (b) 6x +2 (c) 6x-2 (d) 6x-1

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14. If and , then fg(-3) =(a) 45 (b) -45 (c) 33 (d) -33

15. If is the domain of the function then the range is (a) (b) (c) -1 (d) < x <3

16. The following diagram is applicable to question 16 -18

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A possible equation of the graph above is(a) (b) (c) (d)

17. The above graph has this number of roots(a) 1 (b) 2 (c) 0 (d) 3

18. The y- intercept of the graph is (a) -2 (b) 3 (c) 0 (d) -6

19.x 1 2 3 4y 5 7 9 11

Given the table above the relation may be mapped by the relation,(a) y = 2x +1 (b) y = 3x -1 (c) y = 2x +3 (d) y = 2x -3

20. Given that . Then (a) 4x (b) 4x2 (c) 8x (d) 8x2

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