COMPOSITION ON FUNCTIONllllllllllll.pptx

Introduction

In mathematics, functions are fundamental building blocks used to describe relationships between variables. A function can be defined as a rule or mapping that assigns exactly one output value to each input value. Functions can model various real-world situations, such as population growth, temperature changes, or financial trends. One of the fundamental operations that can be performed on functions is addition. Adding functions combines the output values of two functions to form a new function, capturing the combined effect of both original functions.

Definition of Function Addition

The addition of functions involves creating a new function whose output value is the sum of the output values of two individual functions. Mathematically, if we have two functions and , their sum is denoted as:



This notation indicates that the value of the new function at any input is obtained by adding the corresponding values of and at .

Example of Function Addition

Suppose we have two functions:



The sum of these functions is:



Substitute the expressions for and :



Simplify:



Thus, the new function formed by the addition of and is:



Properties of Addition of Functions

1. Commutative Property

The addition of functions satisfies the commutative property, which means the order of addition does not matter:



2. Associative Property

Function addition also satisfies the associative property, meaning when adding three functions, the grouping of the functions does not affect the result:



3. Identity Function

If there is a function defined as (a constant zero function), adding this function to any function does not change the function:



4. Closure Property

The sum of two functions is always a function itself. If and are functions, then is also a function.

Graphical Interpretation

Graphically, the addition of functions can be visualized by adding the corresponding -coordinates of the graphs of and at each -value.

Suppose the graph of is a line and the graph of is a parabola. The graph of would combine the vertical displacements of both graphs at each point along the -axis. This new graph represents the sum function.

Real-World Applications of Function Addition

Economics: Combining revenue and cost functions to calculate profit.

Physics: Adding forces acting on an object to find the net force.

Engineering: Combining electrical signals to find the resultant signal.

Biology: Summing population growth rates from different sources.

Practice Problems

Given:



Find:



Given:



Find:



Conclusion

The addition of functions is a fundamental operation in mathematics that combines the effects of two functions to produce a new function. Understanding how to add functions and interpret their results is essential in various fields such as science, economics, and engineering. Mastering this concept lays the groundwork for more complex operations with functions, such as multiplication, division, and composition.

1. COMPOSITION ON FUNCTIONS

2. I can define composition; and 2. I can obtain a composite of functions. objectives

3. the manner in which the parts of a thing are put together  com·​po·​ si ·​ tion

4. COMPOSITION ON FUNCTIONS Let f and g be functions. The composite function denoted by ( 𝑓 ° 𝑔 )( 𝑥 ) = 𝑓 ( 𝑔 ( 𝑥 )). The process of obtaining a composite function is called function composition.

5. Example 1. Given the functions: 𝑓 ( 𝑥 ) = 𝑔 ( 𝑥 ) = 2x , Find ( 𝑓 ° 𝑔 )( 𝑥 )   Solution: a. ( 𝑓 ° 𝑔 )( 𝑥 ) = 𝑓 ( 𝑔 ( 𝑥 )) definition of composition of functions b. ( 𝑓 ° 𝑔 )( 𝑥 ) = c. ( 𝑓 ° 𝑔 )( 𝑥 ) = d. ( 𝑓 ° 𝑔 )( 𝑥 ) = e. ( 𝑓 ° 𝑔 )( 𝑥 ) =  

6. 2. Given the functions: 𝑓 ( 𝑥 ) = 𝑔 ( 𝑥 ) = , Find ( 𝑓 ° g)( x )   Solution: a. ( 𝑓 ° 𝑔 )( 𝑥 ) = 𝑓 ( 𝑔 ( 𝑥 )) definition of composition of functions b. ( 𝑓 ° 𝑔 )( 𝑥 ) = 2x + 2 c. ( 𝑓 ° 𝑔 )( 𝑥 ) = d. ( 𝑓 ° 𝑔 )( 𝑥 ) = e. ( 𝑓 ° 𝑔 )( 𝑥 ) =  

7. Let’s Try! 1. Given f(x) = ; g(x) = 3x, find f ° g (x).  

8. Let’s Try! 2 . Given f(x) = ; g(x) = x+1 , find f ° g (x).  

9. Let’s Try! 3 . Given f(x) = ; g(x) = x+3 , find f ° g (x).  

10. Assessment: Directions: Find (f ° g)(x) using the two functions given in each number. 1) f (x) = 5x -10 g(x) = x -2   2) f(x) = x + 6 g(x) = 6x 2   3) f(x) = x 2 g(x) = x+2