Math: Domain, Range, and Graphs of Functions
Finding domain and range from equations, graphs, and tables
Math: Domain, Range, and Graphs of Functions
Finding domain and range from equations, graphs, and tables
Math - Grade 9-12
- 1
Find the domain and range of the function y = 2x + 5.
Linear functions with no denominators or square roots usually have no restrictions.
The domain is all real numbers because any real value of x can be substituted into the equation. The range is all real numbers because the linear function can produce any real y-value. - 2
Find the domain of the function y = 1 / (x - 4).
The domain is all real numbers except x = 4 because division by zero is undefined. - 3
Find the domain of the function y = square root of (x + 7).
Set the radicand greater than or equal to 0.
The domain is x greater than or equal to -7 because the expression inside a square root must be nonnegative. - 4
Find the range of the function y = x^2 - 9.
The range is y greater than or equal to -9 because x^2 is always at least 0, so the smallest value of y occurs at x = 0 and is -9. - 5
A function has the set of ordered pairs {(1, 3), (2, 5), (4, 5), (7, -1)}. State the domain and range.
Domain comes from x-values and range comes from y-values.
The domain is {1, 2, 4, 7} because those are the x-values. The range is {-1, 3, 5} because those are the y-values, with no repeats listed. - 6
The table below shows a function: x-values: -2, 0, 3, 5 and y-values: 4, 1, 4, 9. State the domain and range.
The domain is {-2, 0, 3, 5}. The range is {1, 4, 9} because 4 is only listed once in the set of outputs. - 7
Determine whether the relation {(0, 2), (1, 4), (1, 5), (3, 7)} is a function. Explain your answer.
A function cannot assign two different outputs to the same input.
This relation is not a function because the input x = 1 is paired with two different outputs, 4 and 5. - 8
Find the domain of the function y = square root of (5 - x).
The domain is x less than or equal to 5 because the expression inside the square root must be nonnegative, so 5 - x must be at least 0. - 9
Find the range of the function y = absolute value of x + 2.
The smallest value of absolute value of x is 0.
The range is y greater than or equal to 2 because absolute value outputs are always nonnegative, and adding 2 shifts the minimum value up to 2. - 10
A graph is a parabola opening upward with vertex at (3, -4). State the domain and range.
The domain is all real numbers because the parabola extends left and right without end. The range is y greater than or equal to -4 because the vertex gives the minimum y-value. - 11
A graph is a horizontal line at y = -6. State the domain and range.
The domain is all real numbers because the line extends forever to the left and right. The range is {-6} because every point on the graph has y-value -6. - 12
Find the domain of the function y = (x + 1) / (x^2 - 9).
Factor the denominator first.
The domain is all real numbers except x = -3 and x = 3 because those values make the denominator equal to zero. - 13
Find the range of the function y = -x^2 + 1.
The range is y less than or equal to 1 because the parabola opens downward and has a maximum value of 1 at x = 0. - 14
A graph begins at the point (2, 1) and continues to the right and upward like a square root graph. State the domain and range.
Use the starting point to identify the smallest x-value and smallest y-value.
The domain is x greater than or equal to 2 because the graph starts at x = 2 and moves right. The range is y greater than or equal to 1 because the graph starts at y = 1 and moves upward. - 15
Write a possible function that has domain x greater than or equal to 0 and range y greater than or equal to -5.
One possible function is y = square root of x - 5. Its domain is x greater than or equal to 0 because x is inside a square root, and its range is y greater than or equal to -5 because the square root output starts at 0 and is shifted down 5 units.