COMPOSITION OF FUNCTIONS

WHAT IS ‘COMPOSITION OF FUNCTIONS’ ?

Composition of Functions is the process of combining two functions where one function is performed first and the result of which is substituted in place of each x in the other function.

Given the two functions, f and g, the composition of f with g, denoted by f o g (read as “f circle g”), is defined by the equation:

(f o g) (x) = f [g(x)]wherein f is considered to be the dependent function and g is considered to be the independent function.

MORE ABOUT COMPOSITION OF FUNCTIONS…

Composition of functions is not commutative. f[g(x)] is generally not equal to g[f(x)]

For example, consider f(x) = 2x and g(x) = x - 3.f[g(x)] = 2(x - 3) = 2x - 6g[f(x)] = (2x) - 3 = 2x - 3f[g(x)] is not equal to g[f(x)].

HOW DO WE GET THE COMPOSITION OF FUNCTIONS?

Example: Find the composition of f with g, when f(x) = 5x-3 and g(x) = 3-2x.

SOLUTION:

1. Establish the equation based on the given functions and based on the definition.

-given functions: f(x) = 5x-3 and g(x) = 3-2x

-the definition: (f o g) (x) = f [g(x)]2. Determine the dependent and independent

functions.*dependent function: f*independent function: g

3. Substitute the right of the equation by the given expression of the dependent function.

(f o g) (x) = 5x-3

4. Substitute all x’s of the dependent function by the expression of the independent function.

(f o g) (x) = 5(3-2x)-3

5. Simplify the resulting expresion.(f o g) (x) = 5(3-2x)-3

= 15-10x-3 = -10x+12

Therefore, the composition of f with g is -10x+12 .

NOW, WHAT IF WE ARE ASK TO DO THE

REVERSE OF THE GIVEN EXAMPLE?

Don’t worry ‘coz here’s how to do it…

1. Again, we are going to establish the first equation based on the given functions and based on the definition. So, we have

-given function: f(x) = 5x-3 and g(x) = 3-2x

-the definition: (g o f) (x) = g [f(x)]

2. This time:*dependent function: g*independent function: f

3. Substitute the right of the equation by the given expression of the dependent function.

(g o f) (x) = 3-2x

4. Substitute all x’s of the dependent function by the expression of the independent function.

(g o f) (x) = 3-2(5x-3)

5. Simplify the resulting expresion.(g o f) (x) = 3-2(5x-3) = 3-10x+6

= -10x+9

Therefore, the composition of g with f is

-10x+9 .

From the given example wherein we took the composition of f with g (f o g) and the composition of g with f (g o f), we can say that changing the order of functions does not mean that we are going to arrive at the same answer. Thus, changing the order of the functions can result to equal or unequal values of composition.

But, there are instances wherein even if we interchange the order of the functions, their compositions are the same.

CONSIDER THIS EXAMPLE:

Given the functions, f(x)=5x-7 and g(x)= x+7 ,Find f o g and g o f. 5

(f o g)(x)= f[g(x)] (g o f)(x)= g[f(x)] = 5x-7 = x+7 = 5(x+7)-7 5

5 = 5x-7+7 = 5x+35-7 5

5 = 5x = x+7-7 5 = x = x

The pair of functions which arrives at the same value of composition after

interchanging the order is called Inverse Functions.

The composition of inverse functions is always equal to x.

Now let’s see if you can

do it…

Evaluate the composite function f[g(x)] for f(x) = 3x2 + 6 and g(x) = x - 8.

Choices: A. x - 8 B. 3x2 - 48x + 198C. 3x2 - 2D. 3x2 + 6

Correct Answer: B

SOLUTION:

Step 1: f[g(x)] = f[x - 8]

Step 2: = 3(x - 8)2 + 6

Step 3: = 3(x2 - 16x + 64) + 6

Step 4: = 3x2 - 48x + 198

We hope you got it ;)))

Another examples: Let g(x) = 4x2 – 5x and h(x) = x+1

find:a. (g o h)(x)b. (h o g)(x)

Answers:a. 4x2 + 3x – 1b. 4x2 - 5x +1

INVERSE FUNCTIONS

DEFINITIONS OF INVERSE FUNCTIONS:

A function and its inverse function can be described as the "DO" and the "UNDO" functions.  A function takes a starting value, performs some operation on this value, and creates an output answer.  The inverse function takes the output answer, performs some operation on it, and arrives back at the original function's starting value.

This "DO" and "UNDO" process can be stated as a composition of functions.  If functions f and g are inverse functions, then f(g(x))=g(f(x))=x. 

Two functions are inverse if, and only if, every domain of one function can be found on the range of the other and vice-versa, and their composition is always equal to x.

SO HOW DO WE FIND THE INVERSE OF A FUNCTION?

Basically speaking, the process of finding an inverse is simply the swapping of the x and y coordinates.  This newly

formed inverse will be a relation, but may not necessarily be a function. 

Consider this subtle difference in terminology:

Definition:  INVERSE OF A FUNCTION:  The relation formed when the independent variable is exchanged with the dependent variable in a given relation.  (This inverse may NOT be a function.)

Definition:  INVERSE FUNCTION:  If the above mentioned inverse of a function is itself a function, it is then called an inverse function. 

REMEMBER:

The inverse of a function may not always be a function!

The original function must be a one-to-one function to guarantee that its inverse will also be a function. 

CONSIDER THE FOLLOWING EXAMPLES:

IN FINDING THE INVERSE OF A FUNCTION GIVEN AN EQUATION, THE FOLLOWING STEPS CAN BE FOLLOWED:

1. Replace f(x) by y.2. Substitute x with y and y with x.3. Express y as a function of x and

simplify.4. Denote the inverse as g(x). Check the

inverse by applying composition of function. That is:

f(g(x)) = g(f(x)) = x

Find the inverse of the function described by the equation f(x) = 2/3x – 4

SOLUTION:1. Replace f(x) by y.

y = 2/3x -42. Substitute x with y and y with x.

x = 2/3y -43. Express y as a function of x and simplify.

2y = x +4 3 3 ( 2y = x+4) 2 3y = 3x+6 2

That is the inverse of the given equation.

g(f(x)) = 3x + 6 2 = 3 (2x – 4)

+ 6 2 3 = x – 6 + 6 = x

4. Denote the inverse as g(x). Check the inverse by applying composition of function.

f(g(x)) = 3 x – 4 2 = 2 (3x +

6) – 4 3 2 = x + 4 – 4

= x

Since f(g(x)) = g(f(x)) = x , we can say the given equation and the computed inverse are really inverse functions.

TRY THESE! Find the inverse of the function f(x) = x – 4.

Find the inverse of the function    (given that x is not equal to 0).

Find the inverse of the function f(x) = -5x + 4

Answer: y = -x +4 5

Find the inverse of the function g(x) = x +5

Answer: x – 5

Find the inverse of the function h(x) = 5x + 10

Answer: x – 2 5

Thank You for Listening!

(by: Group 1)

Charliez Jane Soriano

Denny Rae Sual

Roland Cabarles

Joshua Cericos

Maria Monica Carbon

Jessa Mae Margallo

Aniemhar Cuadrasal

Hanah Nasifah Ali