General Mathematics - Operations on Functions.pptx

Operations on functions involve applying arithmetic operations (addition, subtraction, multiplication, and division) and other functions (composition) to two or more functions, resulting in a new function

1. Operations on Functions General Mathematics

2. Adding Functions On the example on the right, when we adding functions together, you just simply write down the expressions on each function enclosed with a parentheses where its order must follow to the required operation to perform. Find ↓   Order of expressions on the given operation may interchange.

3. Then distribute the operation to the expression on the right . Make sure to include the operator attached on every term. ↓   Lastly, simplify the expression to acquire the result of the operation. ↓  

4. Subtracting Functions On this operation, the process is the same as adding functions together but now subtracting them instead. Find ↓   Make sure to enclose each expression from each function with parenthesis and also the order of expressions based on the given operation.

5. ↓   ↓   Distributing the subtraction operator on a positive term will result to a negative term, while a subtraction can make a negative term positive. Once the a term with a variable has a coefficient of 1, it is advised to remove the coefficient, if without a variable then leave it as is.

6. Multiplying Functions This operation can be done by firstly by enclosing each expression from each function and write them side by side. Find ↓   Although with the given symbol of the operator, you can write the equation without the operator as long as both of the functions were separated.

7. ↓   ↓   Multiply each term to each term of the other expression. Multiplying variables will result to a increase of power by adding the powers of both terms. Lastly, simplify the expression to acquire the result of the operation.

8. Dividing Functions In dividing functions together just like on the example on the right, you need first to write them in an order of the given operation. Find ↓  

9. Factoring Polynomials Before proceeding on the process of dividing functions, it is necessary to know about factoring expressions since its an important step on performing such operation. Lets say you have this expression: To find its factors, just look for its Grea test Common Factor then divide that on each of the term then → You may find the GCF at the coefficient of the highest term  

10. Express the factors by placing the GCF and the factored expression.   You may check if your factoring is correct by evaluating the expression. ↓ When the evaluated expression is the same as the unfactored expression , the factorization is correct  

11. Examples then Factored Expression:   then ↓ Dividing both negative values will result to a positive value Factored Expression:  

12. Examples Factored Expression When Evaluated:   Now the expression is a trinomial, it is commonly a result of 2 binomial factors. The 1 st term of each factor is a factor of the highest term . The 2 nd terms must be a factor of the 3 rd term of the given expression and a 2 nd term of the expression when added .

13. Examples ↓ Factored Expression   ↓ Factored Expression When Evaluated:   Based on the example on the right, although the given expression is not a trinomial , it may still produce a 2 binomial factors .

14. Examples ↓ These are not the final factors   On trinomials with a coefficient of more than 1 on its highest term, you have to first factor out the expression using the coefficient of the highest term.

15. Examples ↓ This is already the final factors Try to evaluate the factors to be sure   Then further factor out the current 2 nd factor to produce simpler factors.

16. Going back to our example on dividing functions, whats next is to factor out those expressions that can be factored out. The you may look for like expressions or terms that can be cancelled out when nothing can be cancelled out then you have acquired the final answer . ↓  

17. There are situations where you are given with expression that cant be factored out or any of the factored expressions can’t be cancelled . So if that happens, then revert the expressions to their original forms and the whole expression will be as is and the final answer . ↓ This will be your final answer  

18. Composite Functions These operation takes the other function as the value of x of the other one. The function to be treated as the value of x of the other one must be the function after the ∘ symbol. Find ↓  

19. ↓ ↓   Then distribute the coefficient beside the substituted expression and further simplify until the final answer were acquired .

20. Another Example Find ↓ Based on the sequence of operations (PEMDAS) we have to evaluate the one with the exponent. Therefore:  

21. Simplify the expression ↓ This will be your final answer