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
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May 21, 24 08:51 AM
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