Find the domain of the function
g(x) = (x+7) / (x^2 - 36)
The domain of a polynomial is R (real).
The domain of a rational function (polynomial divided by another) is R less the points where the denominator becomes zero.
The given function g(x) is a rational function where the denominator can be factorized into
(x^2-36)=(x+6)(x-6)
which enables you to locate the zeroes of the denominator.
The domain of g(x) is R (real) less the two zeroes of the denominator.
To find the domain of a function, we need to consider the values of x for which the function is defined. In this case, the function is defined for all values of x except those that make the denominator equal to zero.
Let's find the values of x that make the denominator zero by setting it equal to zero and solving for x:
x^2 - 36 = 0
This equation can be factored as the difference of squares:
(x + 6)(x - 6) = 0
To solve for x, we set each factor equal to zero:
x + 6 = 0 or x - 6 = 0
Solving each equation for x, we find:
x = -6 or x = 6
So, the values x = -6 and x = 6 make the denominator zero and result in an undefined function. Therefore, the domain of the function g(x) is all values of x except -6 and 6.
In interval notation, the domain of the function g(x) is (-∞, -6) U (-6, 6) U (6, ∞).