f(x)=square root of (5-x)/(x-8)
-What is the domain?
Domain of numerator (5-x): &8477;
Domain of denominator (x-8) : &8477;
However, when x=8, the denominator becomes zero and the function is undefined, therefore the domain of the function is dom f(x): &8477;\8, or
dom f(x): (-∞,8)∪(8,∞)
Domain of numerator (5-x): ℝ
Domain of denominator (x-8) : ℝ
However, when x=8, the denominator becomes zero and the function is undefined, therefore the domain of the function is dom f(x): ℝ\8, or
dom f(x): (-∞,8)∪(8,∞)
To determine the domain of the function f(x) = √((5-x)/(x-8)), we need to identify the values of x for which the function is defined.
The domain of a square root function is restricted by the requirement that the expression inside the square root must be non-negative. In other words, the quantity (5-x)/(x-8) should be greater than or equal to zero.
To find the domain, we need to solve the inequality (5-x)/(x-8) ≥ 0.
1. Determine the critical values by setting the numerator and denominator of the expression equal to zero:
5 - x = 0 and x - 8 = 0
Solving these equations, we find that the critical values are x = 5 and x = 8.
2. Create a sign chart or number line to determine the intervals where the expression is positive or negative.
Choose test values in each interval and substitute them into the inequality (5-x)/(x-8) ≥ 0.
Taking test values less than 5 (e.g., x = 0), between 5 and 8 (e.g., x = 6), and greater than 8 (e.g., x = 10), we can determine the sign of the expression in each interval.
- If x < 5, then both the numerator and denominator are negative, leading to a positive expression ((-)(-) = (+)).
- If 5 < x < 8, then the numerator is positive and the denominator is negative, resulting in a negative expression ((+)(-) = (-)).
- If x > 8, then both the numerator and denominator are positive, leading to a positive expression ((+)(+) = (+)).
3. Check the sign chart or number line and identify the intervals where the expression (5-x)/(x-8) is non-negative (positive or zero). These intervals will determine the domain of the function.
Based on the sign chart, we see that the expression (5-x)/(x-8) is non-negative (positive or zero) for x ≤ 5 and x > 8.
Therefore, the domain of the function f(x) = √((5-x)/(x-8)) is x ≤ 5 and x > 8.