Composition and inverse 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 - 6
g[f(x)] = (2x) - 3 = 2x - 3
f[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 + 198
C. 3x2 - 2
D. 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 – 1
b. 4x2 - 5x +1

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.

Composition and inverse of functions

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 -4
2. Substitute x with y and y with x.
x = 2/3y -4
3. Express y as a function of x and simplify.
2y = x +4
3
3 ( 2y = x+4)
2 3
y = 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

(by: Group 1)
Charliez Jane Soriano
Denny Rae Sual
Roland Cabarles
Joshua Cericos
Maria Monica Carbon
Jessa Mae Margallo
Aniemhar Cuadrasal
Hanah Nasifah Ali

Composition and inverse of functions

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