Functions, Domain, and Range Practice Questions

To review these concepts, go to Functions, Domain, and Range Study Guide.

Functions, Domain, and Range Practice Questions

Problems

Practice 1

Identify whether each equation is a function

1. y = 100x
2. y = 5
3. x = 4
4. y = 4y
5. |y| = x

Practice 2

Use the vertical line test to identify whether each graph is a function.

Practice 3

Find the domain and range of each equation.

1. y = x-12
2. y = x³
3. y = |x|
4. 
5. y = x² + 3

Solutions

Practice 1

1. Every value that can be substituted for x in the equation y = 100x will give us one y value and one y value only, so y = 100x is a function.
2. In the equation y = 5, the value of x does not matter. The value of y will always be 5, which means that for any x value, there is exactly one y value. Because there can never be more than one y value for any x value, y = 5 is a function.
3. In the equation x = 4, only one value can be substituted for x, and that is 4. Only 4 is equal to 4. When x = 4, y has many different values, so x = 4 is not a function.
4. The equation x = 4y is a little easier to understand when it is rewritten in y = form. Divide both sides of the equation by 4: x = 4y is the same as y = . Every value that can be substituted for x in this equation will give us one y value and one y value only, so x = 4y is a function.
5. The equation | y | = x means that the absolute value of y is equal to x. When the absolute value of a number is taken, any negative sign is removed from the number. The absolute value of 3 and the absolute value of –3 are both 3. This means that every positive value of x has two y values, a positive value and a negative value, so the equation | y | = x is not a function.

Practice 2

1. The following graph passes the vertical line test, because a vertical line can be drawn anywhere and it will cross the graph in no more than one place. This graph is a function.

   

2. The following graph does not pass the vertical line test, because a vertical line can be drawn anywhere on the right side of the y-axis and it will cross the graph in two places. This graph is not a function.

   

3. This graph passes the vertical line test, because a vertical line can be drawn anywhere and it will cross the graph in no more than one place. This graph is a function.

   

4. The following graph does not pass the vertical line test, because a vertical line can be drawn anywhere between x = –4 and x = 4 and it will cross the graph in three places. This graph is not a function.

   

Practice 3

1. Any real number can be substituted for x in the equation y = x – 12, so the domain of the equation is all real numbers. The values that we get back for y are also real numbers, so the range of the equation is all real numbers.
2. Any real number can be substituted for x in the equation y = x³, so the domain of the equation is all real numbers. The values that we get back for y are also real numbers, including negative numbers, so the range of the equation is all real numbers.
3. Any real number can be substituted for x in the equation y = | x | , so the domain of the equation is all real numbers. However, if a number is negative, taking the absolute value of the number removes the negative sign. The only values we can get back for y are zero and positive real numbers, so the range of the equation is zero and all positive real numbers.
4. The denominator of a fraction cannot be equal to zero, so we can substitute any real number for x in the equation except 7, because 7 would make the fraction undefined. The domain of the equation is all real numbers except 7. When these values are put into the equation, we can get back any y value except 0. The range of the equation is all real numbers except 0.
5. Any real number can be substituted for x in the equation y = x² + 3, so the domain of the equation is all real numbers. Because x is squared, the values that we get back for y are only positive real numbers. If x = 0, then x² = 0, and y = 3. Any other value for x will cause a number greater than 0 to be added to 3, which would make y greater than 3. The smallest possible y value is 3. The range of the equation is all real numbers greater than or equal to 3.