3_Operations_on_Functions...........pptx

general mathematics

1. Operations of Functions General Mathematics

2. Lesson Objectives At the end of the lesson, the students must be able to: f ind the sum of functions; d etermine the difference between functions; i dentify the product of functions; f ind the quotient between functions; and determine the composite of a function.

3. Addition of Functions Let f and g be any two functions . The sum f + g is a function whose domains are the set of all real numbers common to the domain of f and g , and defined as follows : ( f + g)(x) = f(x) + g(x)

4. Example 1 If f (x) = 3x – 2 and g (x) = x 2 + 2x – 3, find ( f + g) (x ). Solution to Example 1 ( f + g) (x) = f (x) + g (x ) = (3x – 2) + ( x 2 + 2x – 3 ) = x 2 + 5x – 5

5. Subtraction of Functions Let f and g be any two functions . The difference f – g is a function whose domains are the set of all real numbers common to the domain of f and g , and defined as follows : (f – g)(x) = f(x) – g(x)

6. Example 2 Let f (x) = x 2 – 5 and g (x) = 5x + 4, find (f – g)(x ). Solution to Example 2 (f – g)(x) = f (x) – g (x) = (x 2 – 5) – (5x + 4) = x 2 – 5 – 5x – 4 = x 2 – 5x – 9

7. Multiplication of Functions Let f and g be any two functions . The product fg is a function whose domains are the set of all real numbers common to the domain of f and g , and defined as follows : ( fg )(x) = f(x) · g(x)

8. Example 3 If f (x) = 3x – 2 and g (x) = x 2 + 2x – 3, find ( fg ) (x ). Solution to Example 3 (fg)(x) = (3x – 2)(x 2 + 2x – 3 ) = 3x (x 2 + 2x – 3) – 2(x 2 + 2x – 3) = 3x 3 + 6x 2 – 9x – 2x 2 – 4x + 6 = 3x 3 + 4x 2 – 13x + 6

9. Division of Functions Let f and g be any two functions . The quotient f/g is a function whose domains are the set of all real numbers common to the domain of f and g , and defined as follows : , where g(x) ≠ 0.

10. Example 4 If f (x) = x + 3 and g (x) = x 2 + 2x – 3, find ( f/g ) (x ). Solution to Example 4

11. Composition of Functions The composition of the function f with g is denoted by and is defined by the equation: The domain of the composition function f  g is the set of all x such that x is in the domain of g; and g (x) is in the domain of f.

12. Example 5 Given f(x) = 4x – 5 and g(x) = x2 + 4, find . Solution to Example 5

13. Exercise A Determine whether or not each statement is True or False . If f(x) = x – 3 and g(x) = x + 4, then (f – g)(x) = –7 . If f(x) = 4x – 12 and g(x) = x – 3, then (f + g)(2) = –5 . If f(x) = x + 3 and g(x) = 4x, then (f · g)(2) = 40 . If f(x) = x + 6 and g(x) = 3x, then (f/g)( 3) = 1.

14. Exercise B Find f + g , f – g , fg , and f/g. If f(x) = x – 3 and g (x) = x + 4, f (x) = 3x + 4, g(x) = 2x – 1 f (x) = 2x – 5, g(x) = 4x 2 f (x) = x – 1, g(x) = 2x 2 + x – 3 , 5. ,