By LearningExpress Editors

Updated on Oct 3, 2011

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

*y*= 100*x**y*= 5*x*= 4*y*= 4*y*- |
*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.

*y*=*x*-12*y*=*x*^{3}*y*= |*x*|*y*=*x*^{2}+ 3

**Solutions**

**Practice 1**

- Every value that can be substituted for
*x*in the equation*y*= 100*x*will give us one*y*value and one*y*value only, so*y*= 100*x*is a function. - 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. - 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. - The equation
*x*= 4*y*is a little easier to understand when it is rewritten in*y*= form. Divide both sides of the equation by 4:*x*= 4*y*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*= 4*y*is a function. - 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**

- 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.
- 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. - 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.
- 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**

- 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. - Any real number can be substituted for
*x*in the equation*y*=*x*^{3}, 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. - 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. - 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. - Any real number can be substituted for
*x*in the equation*y*=*x*^{2}+ 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*x*^{2}= 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.

From Algebra in 15 Minutues A Day. Copyright © 2009 by LearningExpress, LLC. All Rights Reserved.

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