Domain and Range

Show your work for each problem.

1.
What does the domain of a function represent?
1. a)The highest point on the graph
2. b)All possible input values (x-values)
3. c)The slope of the function
4. d)All possible output values (y-values)
Answer: All possible input values (x-values)
The domain represents all possible input values (x-values) that a function can accept. The range represents the output values (y-values).
2.
A graph exists from to . What is the domain in interval notation?
1. a)
2. b)
3. c)
4. d)
Answer:
Since the graph exists FROM TO , both endpoints are included. We use square brackets to show inclusion: .
3.
A parabola has its vertex at and opens upward. What is the minimum value of the range?
Answer: 3
When a parabola opens upward, the vertex is the minimum point. The y-coordinate of the vertex is , so the minimum value of the range is .
4.
For , what value of is NOT in the domain?
Answer: 0
In the function , the denominator is . When , we get , which is undefined. Therefore, is not in the domain.
5.
Which bracket notation should you always use with infinity ()?
1. a)Curly bracket
2. b)Either one works
3. c)Square bracket
4. d)Parenthesis
Answer: Parenthesis
Always use parentheses with infinity because infinity is not an actual number that can be reached or included. For example: or .
6.
Find the domain of
Answer: (-∞, 5) ∪ (5, ∞)
1. What operation creates the restriction in this function? division
2. Set the denominator equal to zero: . Solve for x. 5
3. This value must be excluded. Write using union notation. (-infinity, 5) U (5, infinity)
7.
Find the domain of
Answer: [4, ∞)
1. What must be true about the expression under a square root? greater than or equal to 0
2. Set up the inequality: . Solve for x. x >= 4
3. Write the domain in interval notation. [4, infinity)
8.
The range of a function is . What is the minimum y-value the function can produce?
Answer: -2
In the interval , the square bracket at means it is included. So the minimum y-value is , and the function can produce any value from upward.
9.
What is the domain of ?
1. a)
2. b)
3. c)
4. d)
Answer:
The function is a polynomial. Polynomials have no restrictions - no fractions (no division by zero possible) and no square roots. Any real number can be input, so the domain is .
10.
Find the domain and range of
Answer: Domain: (-∞, ∞), Range: [2, ∞)
1. Are there any restrictions on the domain (fractions or square roots)? no
2. What is the domain? (-infinity, infinity)
3. Since for all x, what is the minimum value of ? 2
4. What is the range? [2, infinity)
11.
For , what is the range in interval notation?
Answer: [0, ∞)
The square root function always produces non-negative outputs. When , . As x increases, increases without bound. Range: .
12.
Find the domain of
Answer: (-∞, -3) ∪ (-3, 3) ∪ (3, ∞)
1. Set the denominator equal to zero: x^2 = 9
2. Solve for x (remember both positive and negative roots) x = 3 or x = -3
3. How many values are excluded from the domain? 2
4. Write the domain using union notation (-infinity, -3) U (-3, 3) U (3, infinity)
13.
What is the domain of ?
1. a)
2. b)
3. c)
4. d)
Answer:
For , we need . Solving: , so . Domain: . Note: is a common error from forgetting to divide by 2.
14.
Find the domain of
Answer: [0, 4) ∪ (4, ∞)
1. What restriction does the square root create? x >= 0
2. What restriction does the denominator create? x != 4
3. Combine both restrictions. x must be... x >= 0 and x != 4
4. Write the final domain in interval notation [0, 4) U (4, infinity)
15.
A parabola opens downward with vertex at . What is the maximum value in its range?
Answer: 8
A downward-opening parabola has its maximum at the vertex. The vertex is , so the y-coordinate is the maximum output. Range: .
16.
Find the range of
Answer: (-∞, 9]
1. Does this parabola open up or down? down
2. Find the x-coordinate of the vertex using where , 2
3. Find the y-coordinate by plugging x = 2 into the function 9
4. Since the parabola opens down, what is the range? (-infinity, 9]