FINDING ARITHMETIC COMBINATIONS OF FUNCTIONS

Finding Arithmetic Combinations of Functions

Find

a) (f + g)(x)

b) (f - g)(x)

c) (fg)(x) and

d) (f/g)(x).

What is the domain of f/g?

Problem 1 :

f(x) = x + 3, g(x) = x - 3

Solution :

a.

(f + g)(x) = f(x) + g(x)

(f + g)(x) = x + 3 + x - 3

(f + g)(x) = 2x

b.  

(f - g)(x) = f(x) - g(x)

(f - g)(x) = x + 3 - x + 3

(f - g)(x) = 6

c.  

(fg)(x) = f(x) × g(x)

(fg)(x) = (x + 3) × (x - 3)

(fg)(x) = x² - 9

d.  

(f/g)(x) = f(x) / g(x)

(f/g)(x) = (x + 3) / (x - 3)

Domain : All real values except 3

Problem 2 :

f(x) = 2x - 5, g(x) = 1 - x

Solution :

a.

(f + g)(x) = f(x) + g(x)

(f + g)(x) = 2x - 5 + 1 - x

(f + g)(x) = x - 4

b.

(f - g)(x) = f(x) - g(x)

(f - g)(x) = 2x - 5 - 1 + x

= 3x - 6

(f - g)(x) = 3(x - 2)

c.

(fg)(x) = f(x) × g(x)

(fg)(x) = (2x - 5) (1 - x)

= 2x - 5 - 2x² + 5x

(fg)(x) = 2x² - 7x + 5              

d.

(f/g)(x) = f(x) / g(x)

(f/g)(x) = (2x - 5) / (1 - x)

Domain: All real values except 1

Problem 3 :

f(x) = 3x², g(x) = 6 - 5x

Solution :

a.

(f + g)(x) = f(x) + g(x)

= 3x² + 6 - 5x

(f + g)(x) = 3x² - 5x + 6

b.

(f - g)(x) = f(x) - g(x)

= 3x² - 6 + 5x

(f - g)(x) = 3x² + 5x - 6

c.

(fg)(x) = f(x) × g(x)

= (3x²) (6 - 5x)

(fg)(x) = 18x² - 15x³

d.

(f/g)(x) = f(x) / g(x)

(f/g)(x) = 3x² / 6 - 5x

Domain: All real values except 6/5

Problem 4 :

f(x) = 2x + 5, g(x) = x² - 9

Solution :

a.

(f + g)(x) = f(x) + g(x)

(f + g)(x) = 2x + 5 + x² - 9

(f + g)(x) = x² + 2x - 4

b.

(f - g)(x) = f(x) - g(x)

(f - g)(x) = 2x + 5 - x² + 9

= -x² + 2x + 14

(f - g)(x) = x² - 2x - 14

c.

(fg)(x) = f(x) × g(x)

(fg)(x) = (2x + 5) (x² - 9)

= 2x³ - 18x + 5x² - 45

(fg)(x) = 2x³ + 5x² - 18x - 45

d.

(f/g)(x) = f(x) / g(x)

(f/g)(x) = (2x + 5) / (x² - 9)

Domain: (-∞, -3),(-3, 3) U (3, ∞)

Problem 5 :

f(x) = x² + 5, g(x) = √1 - x

Solution :

a.

(f + g)(x) = f(x) + g(x)

(f + g)(x) = x² + 5 + √1 - x

b.

(f - g)(x) = f(x) - g(x)

(f - g)(x) = x² + 5 - √1 - x

c.

(fg)(x) = f(x) × g(x)

(fg)(x) = (x² + 5) √1 - x

d.

(f/g)(x) = f(x) / g(x)

(f/g)(x) = (x² + 5) / √1 - x

(f/g)(x) = (x² + 5)√1 - x / (1 - x)

Domain: All real values except 1

Problem 6 :

f(x) = √x² - 4, g(x) = x² / x² + 1

Solution :

a.

(f + g)(x) = f(x) + g(x)

(f + g)(x) = √x² - 4 + x² / (x² + 1)

b.

(f - g)(x) = f(x) - g(x)

(f - g)(x) = √x² - 4 - x² / (x² + 1)

c.

(fg)(x) = f(x) × g(x)

(fg)(x) = (√x² - 4) (x² / (x² + 1))

d.

(f/g)(x) = f(x) / g(x)

(f/g)(x) = (√x² - 4) / (x² / (x² + 1))

Domain : All real values

Problem 7 :

f(x) = 1/x, g(x) = 1/x²

Solution:

a.

(f + g)(x) = f(x) + g(x)

(f + g)(x) = 1/x + 1/x²

(f + g)(x) = (x + 1) / x²

b.

(f - g)(x) = f(x) - g(x)

(f - g)(x) = 1/x - 1/x²

(f - g)(x) = (x - 1) / x²

c.

(fg)(x) = f(x) × g(x)

(fg)(x) = (1/x) (1/x²)

(fg)(x) = 1/x³

d.

(f/g)(x) = f(x) / g(x)

(f/g)(x) = (1/x) / (1/x²)

(f/g)(x) = x

Domain : All real values

Problem 8 :

f(x) = x/x + 1, g(x) = 1/x³

Solution :

a.

(f + g)(x) = f(x) + g(x)

(f + g)(x) = x/(x + 1) + 1/x³

(f + g)(x) = (x⁴ + x + 1) / (x⁴ + x³)

b.

(f - g)(x) = f(x) - g(x)

(f - g)(x) = x/(x + 1) - 1/x³

(f - g)(x) = (x⁴ - x - 1) / (x⁴ + x³)

c.

(fg)(x) = f(x) × g(x)

(fg)(x) = (x/(x + 1)) (1/x³)

(fg)(x) = 1/(x³ + x²)

d.

(f/g)(x) = f(x) / g(x)

(f/g)(x) = (x/(x + 1)) / (1/x³)

(f/g)(x) = x⁴ / (x + 1)

Domain: All real values except -1