The above function is given:
f(x)= sqrt of (2x-6)/(x-8)
- Sarah describes the domain as [3, ∞).
- Describe why her domain is incorrect.
- Provide the correct domain for the function.
(I sort of understand this).
Sarah is correct in concluding that the domain of √(2x-3) is [3, ∞). But
x-8 = 0 when x=8
division by zero is not defined
8 is not in the domain, so [3, ∞) cannot be the domain, since it includes 8.
So, what is the domain?
I think the domain would be (-∞, 7].
I am not sure if it is correct. I got the above part correct though.
No, the domain is
[3,8)U(8,∞)
To determine the correct domain of a function, we need to consider any restrictions on the input values that would cause the function to be undefined. In this case, the given function is a square root function:
f(x) = sqrt((2x-6)/(x-8))
For the square root function to be defined, the expression inside the square root must be non-negative (greater than or equal to zero). Here's how you can determine the correct domain:
1. Set the expression inside the square root greater than or equal to zero:
2x-6/(x-8) ≥ 0
2. Find the critical values of x that make the expression equal to zero.
To find the critical values, set the numerator equal to zero:
2x-6 = 0
2x = 6
x = 3
However, we also need to check if the denominator (x-8) is equal to zero, as that would make the function undefined. Setting the denominator equal to zero:
x - 8 = 0
x = 8
So, the critical values are x = 3 and x = 8.
3. Use these critical values to divide the number line into intervals.
We have three intervals to consider: (-∞, 3), (3, 8), and (8, ∞).
4. Test a value from each interval in the original inequality.
Starting with the first interval, choose a test value less than 3; let's say x = 0.
Plugging this value into the original inequality:
(2x-6)/(x-8) = (2(0)-6)/(0-8) = -6/-8 = 3/4
Since 3/4 is positive, the expression inside the square root is greater than zero for x = 0. Therefore, the first interval (-∞, 3) is part of the domain.
Next, choose a test value between 3 and 8; let's say x = 5:
(2x-6)/(x-8) = (2(5)-6)/(5-8) = 4/-3
Since the expression is negative for x = 5, the second interval (3, 8) is not part of the domain.
Finally, choose a test value greater than 8; let's say x = 10:
(2x-6)/(x-8) = (2(10)-6)/(10-8) = 14/2 = 7
Since 7 is positive, the third interval (8, ∞) is part of the domain.
5. Combine the intervals that satisfy the condition.
Based on our tests, the correct domain for the function is the union of the intervals (-∞, 3) and (8, ∞). Therefore, the correct domain for the function is (-∞, 3) U (8, ∞) or simply (-∞, 3) ∪ (8, ∞).