Lesson 3: Operations on FunctionsSay we have two functions called f(x) and g(x). The dierent operations onthese two functions is shown below: Adding two functions as:(f+g)(x) = f(x)+g(x) Subtracting two functions as:(f-g)(x) = f(x) – g(x) Multiplying two functions as:(f g)(x) = f(x) g(c) Dividing two functions as:( f g)(x) = f(x ) g(x ) ; whereg(x ) 6= 0Example 1:Let f(x) = 4x + 5 and g(x) = 3x. Find (f+g)(x), (f-g)(x), (f g)(x), and ( f g)(x). (f+g)(x) = (4x+5) + (3x) = 7x+5 (f-g)(x) = (4x+5) – (3x) = x+5 (f g)(x) = (4x+5) (3x) = 12 x2+5x (f g)(x) = 4x +5 3xExample 2:Let f(x)= 3x+2 and g(x)= 5x-1.
Find (f+g)(x), (f-g)(x), (f g)(x), and ( f g)(x). (f+g)(x) = (3x+2) + (5x-1) = 8x+1 (f-g)(x) = (3x+2) – (5x-1) = -2x+3 (f g) = (3x+2) (5x-1) = 15 x2+7x -2 (f g)(x) = 3x +2 5x 1Example 3:Let v(x) = x3and w(x) = 3 x2+5x. Find (v+w)(x), (v-w)(x), (v w)(x), and( v w)(x). (v+w)(x) = ( x3) + (3 x2+5x) = x3+ 3 x2+5x (v-w)(x) = ( x3) (3×2+5x) = x3 3x 2-5x (v w) = ( x3) (3×2+5x) = 3 x5+ 5 x4 (v w)(x) = ( x3 3x 2+5 x) = xx 2 x(3 x+5) = x2 3x +5Example 4:Let f(x) = 4 x3+ 2 x2+4x + 1 and g(x) = 3 x5+ 4 x2+8x-12.
Find (f+g)(x),(f-g)(x), (f g)(x), and ( f g)(x).1(f+g)(x) = (4 x3+ 2 x2+4x+1) + (3 x5+ 4 x2+8x-12) = 3 x5+ 4 x3+ 6 x2+12x-11 (f-g)(x) = (4 x3+ 2 x2+4x+1) – (3 x5+ 4 x2+8x-12) = 3x 5+ 4 x3 2x 2-4x+13 (f g)(x) = (4 x3+ 2 x2+4x+1) (3 x5+ 4 x2+8x-12)= 12 x8+ 6 x7+ 12 x6+ 19 x5+ 40 x4 16×3+ 12 x2 40x 12 (f g)(x) = (4×3+2 x2+4 x+1) (3×5+4 x2+8 x 12)Example 5:Let h(x) = 1 and g(x) = x4 x3+ x2-1. Find (h+g)(x), (h-g)(x), (h g)(x),and ( h g)(x). (h+g)(x) = (1) + ( x4 x3+ x2-1) = x4 x3+ x2 (h-g)(x) = (1) – ( x4 x3+ x2-1) = x4 x3+ x2+2 (h g)(x) = (1) (x 4 x3+ x2-1) = x4 x3+ x2-1 (h g)(x) = 1 x4 x3+ x2 12Exercises:1. If h(x) = 7x+3 and g(x) = 2 x2+1. Find (f+g)(x)2. If f(x) = x5-18 and g(x) = x2- 6x + 9, what is the vaue of (g-h)(x)?3. If f(x) = 25 x5and g(x) = 55 x8, what is the value of ( f g)(x)?4.
If v(x) = x3and w(x) = x2+ 4, what is the value of (v w)(x)?5. If f(x) = 4x + 11 and g(x) = 5x + 9, nd (f+g)(x).6. If f(z) = 7z – 4 and g(z) = z-2, nd (f-g)(x).7.
If f(x) =8 x2-20 and g(x) =-4, what is the value of ( f g)(x).8. If f(x) = 2x+2 and g(x) = 9 x2, what is the value of (f g)(x)?9.
If f(x) = 7 x2+ 8x -3 and g(x) = 7x, what is the value of (f g)(x)?10. If f(x) = 35 x8- 45x and g(x) = 5x, what is the value of ( f g)(x).3Answers to Operations on Functions Exercises:1. 2 x2+7x +42.
x5 x2+ 6x – 273. 5x 11×34. x5+ 4 x35. 9x +206.
6z -27. 2x 2+ 58. 18 x3+ 18 x29. 49 x3+ 56 x2- 2110.7 x7- 94
