Math help please

Question

Given functions g(x)=1√xg(x)=1x and h(x)=x2−4h(x)=x2-4, state the domains of the following functions using interval notation.

Domain of   g(x)h(x)    g(x)h(x)  : 

Domain of   g(h(x))    g(h(x))  : 

Domain of   h(g(x))    h(g(x))  : 

Answer 1

The domain of a function is the set of input values for which the function is defined.

The domain of g(x) is the set of all real numbers x except for x = 0, because the function is defined as 1/x and division by 0 is not allowed. Therefore, the domain of g(x) can be written as (-∞,0)∪(0,∞).

The domain of h(x) is the set of all real numbers x except for x = 2 and x = -2, because the function is defined as x^2 - 4 and the square root of a negative number is not a real number. Therefore, the domain of h(x) can be written as (-∞,-2)∪(-2,2)∪(2,∞).

For the function g(x)h(x), the domain is the set of all real numbers x except for x = 0 and x = 2 and x = -2. This can be written as (-∞,-2)∪(-2,0)∪(0,2)∪(2,∞).

For the function g(h(x)), the domain is the set of all real numbers x except for x = 2 and x = -2, because these values would make h(x) negative and result in the square root of a negative number, which is not a real number. Therefore, the domain of g(h(x)) can be written as (-∞,-2)∪(-2,2)∪(2,∞).

For the function h(g(x)), the domain is the set of all real numbers x except for x = 0, because this value would make g(x) equal to infinity and result in division by infinity, which is not allowed. Therefore, the domain of h(g(x)) can be written as (-∞,0)∪(0,∞).