Find the domain of each function.
f(x)=Sqareroot of 5-x
f(x)=x^3/x^2-16
To find the domain of each function, we need to determine the values for which the function is defined. In other words, we should identify any restrictions or limitations on the independent variable (x) that would make the function undefined.
1. f(x) = √(5 - x):
To find the domain of this function, we need to consider the square root (√) term. For a square root to be defined, the radicand (the expression inside the square root) must be non-negative.
Thus, we set 5 - x ≥ 0 and solve for x:
5 - x ≥ 0
-x ≥ -5
x ≤ 5
Therefore, the domain of f(x) = √(5 - x) is x ≤ 5.
2. f(x) = x^3 / (x^2 - 16):
To find the domain of this function, we should look for any values of x that could result in division by zero. In this case, the denominator cannot be equal to zero because it would lead to an undefined value for f(x).
Therefore, we set the denominator equal to zero and solve for x:
x^2 - 16 = 0
(x - 4)(x + 4) = 0
x - 4 = 0 or x + 4 = 0
x = 4 or x = -4
Thus, the domain of f(x) = x^3 / (x^2 - 16) is all real numbers except x = 4 and x = -4.