T IP – FTP – UB

INDUSTRIAL MATHEMATICS - I

INDUSTR IAL MATHEMATIC S - I

FUNCTION

WHAT IS FUNCTION ?

• Imagine : playing golf, putting a golfball into the hole.

• A function is transforming an input x into an output y = f(x).

x yff : x y / y=f(x)

f : x y / y=f(x)y=f(x)=x2

WHAT IS FUNCTION ?

• (Try) Which of the following equations is a function ?(a) y = 1 – x2

(b) y = Functions are rules,(c) y = but not all rules are functions.

• Function is a relation between a set of inputs and a set of permissible outputs, with a property that each input is related to exactly one output.

• Function is a mapping or equivalent rule which connected each object in a sets (domain), with a unique value of f(x) from another sets (range/codomain).

DOMAIN, CODOMAIN, RANGE

• If f mapped or related x A to y B, it is :- said that y is a map from x- written as f : x y or y = f(x)

• Sets y B which is map from x A is called range or result area.

f (a) = 1 range R = {1, 2, 3, 4]

f (b) = 2 f (c) = 3 f (d) = 4

DOMAIN, CODOMAIN, RANGE

• Domain = all the input numbers x that a function can process.

• Co-domain = all the numbers in the sets y.

• Range = complete collection of numbers y that correspond to the numbers is the domain.

• y = domain is -1 ≤ x ≤ 1 , range is 0 ≤ y ≤ 1

• y = x3 , -2 ≤ x ≤ 3 range is -8 ≤ y ≤ 27

EXAMPLES

• Define the domain and range for these equations :(a) y = x3 , -2 ≤ x < 3 (b) y = x4

(c) y = , 0 ≤ x ≤ 6

• Let’s say f : R R with f(x-1) = x2 + 5x, define :(a) f(x)(b) f(-3)

OPERATIONS OF FUNCTION

• Operations of function can be a sum, substract, multiply, or divide with the rules are :

• Example : If F(x) = and G(x) = define : a. F+G(x) b. F-G(x) c. F.G(x) d. F/G(x) e. F5

COMPOSITE FUNCTION

• Function composition is the combining operations of two functions sequentially resulting to another function (composite function).

• Function composition is the application of one function to the results of another.

y=f(x) z=g(y)/z=g(f(x)) mapping of x A to z C

is a composition of f and g

written (g o f)(x) = g(f(x))

COMPOSITE FUNCTION

• Composite function is always associative or not commutative.means f o g ≠ g o f

• Example :f : R R and g : R Rf(x) = 3x – 1 and g(x) = 2x2 + 5Define : a. (g o f)(x) and b. (f o g)(x) !

a. (g o f)(x)=g(f(x)) = g(3x – 1)= 2(3x – 1)2 + 5= 2(9x2 – 6x + 1) + 5= 18x2 – 12x + 2 + 5= 18x2 – 12x + 7

b. (f o g)(x) = …..??

COMPOSITE FUNCTION

• How to define a function from a known function composition ?

• Example :Given f(x) = 3x – 1 and (f o g)(x) = x2 + 5, define g(x) !

Answer :(f o g)(x) = x2 + 5f(g(x)) = x2 + 53.g(x) – 1 = x2 + 53.g(x) = x2 + 6

g(x) = 1/3(x2 + 6)

Try Given g(x) = 2x2 + 2 and (g o f)(x) = x – 3 , define f(x) !

INVERSE FUNCTION• Invers function is a function that undoes another function : If an input x into the

function f produces an output y, then putting y into the function g produces the output x g is an invers function of f.

If, f : A B = f : {(a,b,c,1,2,3)|a,b,c A and 1,2,3 B}

Then f-1 : B A = f : {(1,2,3,a,b,c)|1,2,3 B and a,b,c A}

• A function f that has an inverse is called invertible; denoted by f-1.

f : x y or y = f(x)f-1 : y x or x = f-1(y) y = f-1(x)

INVERSE FUNCTION• Inverse Function, another explanation.

INVERSE FUNCTION

• Example : Determine the inverse function from function f(x) = 2x – 6y = f(x) = 2x – 6 y = 2x – 6 2x = y + 6 x = ½(y + 6)So, x = f-1(y) = ½ (y + 6) f-1(x) = ½ (x + 6)

• Now determine the inverses from this function !! :

COMPOSITION AND INVERSE FUNCTION

• How is the function is a combination of composition and invers function ?

Function composition Invers function (reverse way) h = (g o f) h-1 = f-1 o g-1

(g o f)-1 = f-1 o g-1

• Example :If f : R R and g : R R determined by function f(x) = x + 3 and g(x) = 5x – 2 , define (f o g)-1(x) !!

COMPOSITION AND INVERSEFUNCTION

• Example :If f : R R and g : R R determined by function f(x) = x + 3 and g(x) = 5x – 2 , define (f o g)-1(x) !!

Solution 1 = Find (f o g)(x) first, then define (f o g)-1(x)

(f o g)(x) = f(g(x)) = (5x – 2) + 3 y = 5x + 1 5x = y – 1 x = 1/5(y – 1) = 1/5y – 1/5

So, (f o g)-1(x) = 1/5x – 1/5

COMPOSITION AND INVERSEFUNCTION

• Example :If f : R R and g : R R determined by function f(x) = x + 3 and g(x) = 5x – 2 , define (f o g)-1(x) !!

Solution 2 = Find f-1(x) and g-1(x) first,then use (f o g)-1(x) = (g-1 o f-1)(x)

(f o g)-1(x) = (g-1 o f-1)(x) = g-1(f-1(x)) = 1/5(x – 3) + 2/5 = 1/5x – 3/5 + 2/5 = 1/5x –

1/5

f (x) = x + 3 y = x + 3 x = y – 3 f-1(x) = x – 3

g (x) = 5x – 2 y = 5x – 2 x = 1/5 y + 2/5 g-1(x) = 1/5 x + 2/5

TIP APPLICATION

TASK1. If f(x) = 2x + 1 and g(x) = , determine (g o f)-1(x) !

2. If f(x) = and g(x) = 2x – 1 , determine (fog)-1(x) !

3. If , find f-1(1) !

4. f(x) = 2x – 3 , f-1(-1) = …..

5. If f(x) = and (f o g)(x) = 2x – 1 , find g(x) !

6. If f(x) = 2x – 1 for –2 < x < 4 and g(x) = for 3 < x < 5 , find the domain and range of !

7. If f(x+2) = 2x3 – 4x + 3

TASK SCORE

(1). If f(x) = 2x + 1 and g(x) = , determine (g o f)-1(x) !

(gof)(x) = (10)

(gof)-1(x) = (15)ORg-1(x) = (5) f-1(x) = (5) (gof)-1(x) = (f-1 o g-1) (x) = (15)

TASK SCORE

(2). If f(x) = and g(x) = 2x – 1 , determine (fog)-1(x) !

(fog)(x) = (10)(fog)-1(x) = (15)

OR

f-1(x) = (5) g-1(x) = (5) (fog)-1(x) = (g-1 o f-1) = (15)

TASK SCORE

(3). If , find f-1(1) !

(10) (5)

(4). f(x) = 2x – 3 , f-1(-1) = …..

f-1(x) = (10) f-1(-1) = = 1 (5)

TASK SCORE

(5). If f(x) = and (f o g)(x) = 2x – 1 , find g(x) !

(fog)(x) = f(g(x)) = 2x – 1 (5)

g(x) = (5)

(6) (5)

(7) (5)

INDUSTR IAL MATHEMATIC S -1

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