1. industrial mathematics - i
TIP – FTP – UB

2. industrial mathematics - I
function
industrial mathematics - I

3. What is function ? x f y f : x  y / y=f(x) f : x  y / y=f(x)
Imagine : playing golf, putting a golfball into the hole.
A function is transforming an input x into an output y = f(x). x f y
f : x  y / y=f(x)
f : x  y / y=f(x)
y=f(x)=x2

4. What is function ?
(Try) Which of the following equations is a function ?
(a) y = 1 – x2
(b) y = − 𝑥 Functions are rules,
(c) y = 𝑥 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).

5. 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) = range R = {1, 2, 3, 4]
f (b) = 2
f (c) = 3
f (d) = 4

6. 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 = 1− 𝑥 2  domain is -1 ≤ x ≤ 1 , range is 0 ≤ y ≤ 1
y = x3 , -2 ≤ x ≤ 3  range is -8 ≤ y ≤ 27

7. examples Define the domain and range for these equations :
(a) y = x3 , -2 ≤ x < 3
(b) y = x4
(c) y = 1 (𝑥−1)(𝑥+2) , 0 ≤ x ≤ 6
Let’s say f : R  R with f(x-1) = x2 + 5x, define :
(a) f(x)
(b) f(-3)

8. Operations of function
Operations of function can be a sum, substract, multiply, or divide with the rules are :
Example : If F(x) = 4 𝑥+1 and G(x) = 9− 𝑥 2
define : a. F+G(x) b. F-G(x) c. F.G(x) d. F/G(x) e. F5

9. 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))

10. Composite function
Composite function is always associative or not commutative.
means  f o g ≠ g o f
Example :
f : R  R and g : R  R
f(x) = 3x – 1 and g(x) = 2x2 + 5
Define : 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
= 18x2 – 12x + 7
b. (f o g)(x) = …..??

11. 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 + 5
f(g(x)) = x2 + 5
3.g(x) – 1 = x2 + 5
3.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) !

12. 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)

13. Inverse function
Inverse Function, another explanation.

14. InversE function
Example : Determine the inverse function from function f(x) = 2x – 6
y = 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 !! :
𝑓 𝑥 = 3𝑥 −4 2𝑥−1

15. 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) !!

16. Composition and Inverse function
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

17. Composition and Inverse function
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

18. Tip application

19. TASK
1. If f(x) = 2x + 1 and g(x) = 3𝑥+5 𝑥 −4 , determine (g o f)-1(x) ! 2. If f(x) = 𝑥+4 𝑥 −6 and g(x) = 2x – 1 , determine (fog)-1(x) ! 3. If 𝑓 𝑥 = 2𝑥 −5 3𝑥+1 ,𝑥 ≠− 1 3 , find f-1(1) ! 4. f(x) = 2x – 3 , f-1(-1) = ….. 5. If f(x) = 4 𝑥−2 and (f o g)(x) = 2x – 1 , find g(x) ! 6. If f(x) = 2x – 1 for –2 < x < 4 and g(x) = 4 𝑥−2 for 3 < x < 5 , find the domain and range of 2.𝑓(𝑥) 𝑔(𝑥) ! 7. If f(x+2) = 2x3 – 4x + 3

20. TASK score
(1). If f(x) = 2x + 1 and g(x) = 3x+5 x −4 , determine (g o f)-1(x) ! (gof)(x) = 6x+8 2x−3 (10) (gof)-1(x) = 8+3x 2x−6 (15) OR g-1(x) = 5+4x x−3 (5) f-1(x) = 1−x 2 (5) (gof)-1(x) = (f-1 o g-1) (x) = 5+4x x−3 −1 2 = 8+3x 2x−6 (15)

21. TASK score
(2). If f(x) = 𝒙+𝟒 𝒙 −𝟔 and g(x) = 2x – 1 , determine (fog)-1(x) !
(fog)(x) = 2x+3 2x−7 (10)
(fog)-1(x) = 3+7x 2x−2 (15) OR
f-1(x) = 4+6x x−1 (5) g-1(x) = x+1 2 (5) (fog)-1(x) = (g-1 o f-1) = 3+7x 2x−2 (15)

22. TASK score
(3). If f x = 2x −5 3x+1 ,x ≠− 1 3 , find f-1(1) ! f −1 x = −5−x 3x−2 (10)  f −1 1 = −5−1 3−2 = −6 1 =−6 (5) (4). f(x) = 2x – 3 , f-1(-1) = ….. f-1(x) = x+3 2 (10)  f-1(-1) = −1+3 2 = 1 (5)

23. TASK score
(5). If f(x) = 4 x−2 and (f o g)(x) = 2x – 1 , find g(x) ! (fog)(x) = f(g(x)) = 2x – 1  4 g x −2 =2x−1 (5) g(x) = 4x+2 2x−1 (5) (6) (5) (7) (5)

24. Industrial mathematics -1
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Industrial mathematics -1