Please help.
Name all values of x that are not in the domain of the given function.
f(x)=x-10/(sqrt x^2-16)
To find the values of x that are not in the domain of the function f(x), we need to consider any restrictions on the variables in the expression. In this case, we have the function:
f(x) = (x - 10) / (√(x^2 - 16))
The function has two main restrictions:
1. The denominator cannot be zero.
2. The expression under the square root (√) must be non-negative.
Let's analyze these restrictions one by one.
1. The denominator cannot be zero:
To find the values of x that would make the denominator zero, we set it equal to zero and solve for x:
x^2 - 16 = 0
To solve this equation, we factor it as the difference of squares:
(x - 4)(x + 4) = 0
Setting each factor equal to zero, we have:
x - 4 = 0 or x + 4 = 0
Solving for x in each case, we get:
x = 4 or x = -4
Therefore, x cannot be either 4 or -4 since they would make the denominator zero.
2. The expression under the square root must be non-negative:
To ensure that the expression under the square root (√) is non-negative, we need to consider the values of x that would not result in a negative value.
The expression x^2 - 16 represents a square minus a constant. For all values of x except -4 and 4, this quantity is always positive since the square of any real number is non-negative.
So, the values of x that are not in the domain of the function f(x) are:
x = 4 and x = -4
These values would make the denominator zero and thus violate the restrictions of the function.