Find (f + g)x, (f-g)x, (fg)x, and
(f/g)x and their domains.
f(x)=¡î(9-x©÷),g(x)=¡î(x©÷-4)
In western(ISO-8859-1) encoding, the post is:
Find (f + g)x, (f-g)x, (fg)x, and
(f/g)x and their domains.
f(x)=√(9-x²),g(x)=√(x²-4)
When posting mathematical symbols, you have more chances of being understood if you post directly in Western encoding.
========================================
If you are working in the real domain (ℝ), you would want the results of the transformations to remain in ℝ.
For the +,- and * operations, the transformed domain would simply be D(f)∩D(g).
For the division, (f/g)(x), you will need to remove from D(f)∩D(g) points that do not exist in the transformed function. In the case of polynomials, this would generally be the values of x where g(x) becomes zero.
So if you get started with finding the domains of f(x) and g(x), you would have made a big step. All you need to do is to find the domains according to the rules above.
Post your answers for a check if you wish.
To find the expressions (f + g)(x), (f - g)(x), (f * g)(x), and (f / g)(x), we need to perform arithmetic operations on the given functions f(x) and g(x).
1. (f + g)(x):
To find (f + g)(x), substitute f(x) and g(x) into the expression and simplify:
(f + g)(x) = f(x) + g(x)
= √(9 - x²) + √(x² - 4)
The domain of (f + g)(x) is the intersection of the domains of f(x) and g(x), which is the set of real numbers x that satisfy both inequalities:
Domain: {x | 9 - x² ≥ 0} ∩ {x | x² - 4 ≥ 0}
= {x | x² ≤ 9} ∩ {x | x² ≥ 4}
= {x | -3 ≤ x ≤ 3}
Therefore, the domain of (f + g)(x) is [-3, 3].
2. (f - g)(x):
To find (f - g)(x), substitute f(x) and g(x) into the expression and simplify:
(f - g)(x) = f(x) - g(x)
= √(9 - x²) - √(x² - 4)
The domain of (f - g)(x) is the same as the domain of (f + g)(x), which is [-3, 3].
3. (f * g)(x):
To find (f * g)(x), substitute f(x) and g(x) into the expression and simplify:
(f * g)(x) = f(x) * g(x)
= √(9 - x²) * √(x² - 4)
= √((9 - x²) * (x² - 4))
The domain of (f * g)(x) is the intersection of the domains of f(x) and g(x), which is [-3, 3].
4. (f / g)(x):
To find (f / g)(x), substitute f(x) and g(x) into the expression and simplify:
(f / g)(x) = f(x) / g(x)
= (√(9 - x²)) / (√(x² - 4))
The domain of (f / g)(x) is the set of real numbers x that satisfy the condition: x² - 4 ≠ 0 (denominator cannot be zero).
Domain: {x | x² - 4 ≠ 0}
= {x | x ≠ 2, x ≠ -2}
= (-∞, -2) ∪ (-2, 2) ∪ (2, ∞)
Therefore, the domain of (f / g)(x) is (-∞, -2) ∪ (-2, 2) ∪ (2, ∞).