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2. Find the domain of the following function. Your answer should use interval notation. f(x) =

Foundations
The foundations:
What is the domain of g(x) = ?
The function is undefined if the denominator is zero, so x 0.
Rewriting"x 0" in interval notation( -, 0) (0, )
What is the domain of h(x) = ?
The function is undefined if we have a negative number inside the square root, so x 0


Solution:

Step 1:
Factor
So we can rewrite f(x) as
Step 2:
When does the denominator of f(x) = 0?
(x + 1)(x - 2) = 0
(x + 1) = 0 or (x - 2) = 0
x = -1 or x = 2
So, since the function is undefiend when the denominator is zero, x -1 and x 2
Step 3:
What is the domain of
critical points: x = -1, x = 2
Test points:
x = -2: (-2 + 1)(-2 - 2): (-1)(-4) = 4 > 0
x = 0: (0 + 1)(0 - 2) = -2 < 0
x = 3: (3 + 1)(3 - 2): 4*1 = 4 > 0
So the domain of h(x) is
Step 4:
Take the intersection (i.e. common points) of Steps 2 and 3.









2. Find the domain of the following function. Your answer should use interval notation. f(x) =

Foundations
The foundations:
What is the domain of g(x) = ?
The function is undefined if the denominator is zero, so x 0.
Rewriting"x 0" in interval notation( -, 0) (0, )
What is the domain of h(x) = ?
The function is undefined if we have a negative number inside the square root, so x 0


Solution:

Step 1:
Factor
So we can rewrite f(x) as
Step 2:
When does the denominator of f(x) = 0?
(x + 1)(x - 2) = 0
(x + 1) = 0 or (x - 2) = 0
x = -1 or x = 2
So, since the function is undefinend when the denominator is zero, x -1 and x 2
Step 3:
What is the domain of
critical points: x = -1, x = 2
Test points:
x = -2: (-2 + 1)(-2 - 2): (-1)(-4) = 4 > 0
x = 0: (0 + 1)(0 - 2) = -2 < 0
x = 3: (3 + 1)(3 - 2): 4*1 = 4 > 0
So the domain of h(x) is
Step 4:
Take the intersection (i.e. common points) of Steps 2 and 3.









2. Find the domain of the following function. Your answer should use interval notation.

Hint 1
Which x-values lead to division by 0 or square rooting a negative number
Hint 2
Use a sign chart to determine for which x-values

Solution:

Solution
Since the domain is the collection of x-values for which we don't divide by zero or square root a negative number we want to solve the inequality
Now we use a sign chart with test numbers -2, 0, and 3
So the solution is





2. Find the domain of the following function. Your answer should use interval notation.

Hint 1
Which x-values lead to division by 0 or square rooting a negative number
Hint 2
Use a sign chart to determine for which x-values

Solution:

Solution
Since the domain is the collection of x-values for which we don't divide by zero or square root a negative number we want to solve the inequality
Now we use a sign chart with test numbers -2, 0, and 3
So the solution is