1.Find the domain of the function. Make sure answer is in interval notation.
g(x)=7x/x^2-4
2.Find the domain of the function. Make sure answer is in interval notation.
f(x)=5x square root of x-2
1. g(x) = 7x/(x^2-4)
x^2-4 = 0
x^2 = 4
X = +- 2.
Domain: -2 <X< 2.
2. F(x) = 5x*sqrt(x-2).
2 <=X< +Infinity.
Correction:
1. 7x/(x^2-4) = 7x/(x+2)(x-2).
When X = +-2, the denominator = 0. Therefore, +-2 are not in the domain.
Domain: All real values of X except +-2.
To find the domain of a function, we need to determine the values that x can take which will give a valid output. In both cases, we have rational functions, so we need to be careful about avoiding division by zero.
1. Function g(x) = 7x/(x^2 - 4):
The denominator of the function is x^2 - 4, which will be zero when x = ±2. Division by zero is undefined, so we must exclude these values from the domain.
To find the valid domain, we can consider that all real numbers are potential inputs, except for those that make the denominator zero. We exclude x = -2 and x = 2 from the domain.
Therefore, the domain of g(x) is given in interval notation as: (-∞, -2) ∪ (-2, 2) ∪ (2, ∞).
2. Function f(x) = 5x√(x - 2):
In this case, we have a square root term in the denominator. For the square root to be defined, the expression inside it (x - 2) needs to be non-negative (greater than or equal to zero).
The domain will be the set of all x values that make x - 2 ≥ 0. Solving this inequality, we find that x ≥ 2.
Therefore, the domain of f(x) is given in interval notation as: [2, ∞).