Find the domain of the following functions.
1. y= radical(x-3) - radical(x+3)
2. y= [radical(2x-9)] / 2x+9
3. y= radical(x^2 - 5x -14)
4. y= [cubed root(x-6)] / [radical(x^2 - x - 30)]
I'm terrible at finding domains.. so any help with these problems would be appreciated. & if you have any general tips with finding domains, I would love them!
recall that
√z is defined only for z >= 0
division by 0 is undefined
To find the domain of a function, we need to determine the set of all possible values that the independent variable (usually denoted as 'x') can take, for which the function is defined. Here's how we can find the domains of the given functions:
1. y = √(x - 3) - √(x + 3)
For the function to be defined, the expressions inside the square roots must be non-negative. Therefore, we need to solve the inequalities x - 3 ≥ 0 and x + 3 ≥ 0.
Solving x - 3 ≥ 0, we have:
x ≥ 3
Solving x + 3 ≥ 0, we have:
x ≥ -3
Since we want the intersection of both conditions, the domain of the function is:
x ≥ 3
2. y = √(2x - 9) / (2x + 9)
Similar to the previous question, for the function to be defined, the expressions inside the square root and the denominator must be non-negative. First, we need to solve the inequality 2x - 9 ≥ 0.
Solving 2x - 9 ≥ 0, we have:
2x ≥ 9
x ≥ 9/2
Next, we need to solve the inequality 2x + 9 ≠ 0, since division by zero is not allowed. Solving this inequality, we have:
2x ≠ -9
x ≠ -9/2
Taking the intersection of both conditions, the domain of the function is:
x ≥ 9/2, x ≠ -9/2
3. y = √(x^2 - 5x - 14)
In this case, the square root expression must be non-negative for the function to be defined. To determine this, we need to solve the inequality x^2 - 5x - 14 ≥ 0.
Solving x^2 - 5x - 14 ≥ 0, we can either factorize the quadratic or use the quadratic formula. Factoring, we have:
(x - 7)(x + 2) ≥ 0
To solve this inequality, we make a sign chart by considering the intervals between the roots (-2 and 7), and testing a value in each interval.
Interval (-∞, -2):
For x = -3, (x - 7)(x + 2) = (-3 - 7)(-3 + 2) = (-10)(-1) = 10 > 0
Interval (-2, 7):
For x = 0, (x - 7)(x + 2) = (0 - 7)(0 + 2) = (-7)(2) = -14 < 0
Interval (7, ∞):
For x = 8, (x - 7)(x + 2) = (8 - 7)(8 + 2) = (1)(10) = 10 > 0
Since we are interested in the intervals where (x - 7)(x + 2) ≥ 0, the function is defined on the union of the intervals (-∞, -2) and (7, ∞). Therefore, the domain of the function is:
x ∈ (-∞, -2] ∪ [7, ∞)
4. y = ∛(x - 6) / √(x^2 - x - 30)
Similar to the previous functions, we need to ensure that the cube root and the square root expressions are well-defined. First, we solve the inequality x - 6 ≥ 0, which determines the domain for the cube root.
Solving x - 6 ≥ 0, we have:
x ≥ 6
Next, we need to ensure that the square root has a non-negative expression inside. We solve the inequality x^2 - x - 30 ≥ 0.
Solving x^2 - x - 30 ≥ 0, we can again factorize or use the quadratic formula. Factoring, we have:
(x - 6)(x + 5) ≥ 0
Using a sign chart with the intervals (-∞, -5), (-5, 6), and (6, ∞), we find that the function is defined where (x - 6)(x + 5) ≤ 0. Therefore, the domain is given by the intersection of both conditions:
x ≥ 6
Note: It's always a good practice to check for any additional restrictions or limitations in the given functions, such as excluded values due to division by zero or square roots of negative numbers.
General tips for finding domains:
1. Identify any potential restrictions for the function, such as division by zero or square roots of negative numbers.
2. Solve any equations or inequalities to determine the value(s) that will make certain expressions undefined.
3. Combine the conditions to find the intersection of all restrictions, if any.
4. Write the final domain in set notation or interval notation, depending on the context.
Hope this helps! Let me know if you have any other questions.