Relations: Solutions

Introduction to Functions: Solutions

Here are some solutions to the Exercises to accompany the section Introduction to Functions. Some Exercises are designed to be done with a partner and to be graded by the partner.

Consider the Ferris wheel example from the previous section. Use an appropriate test from the text to explain why that relation is not a function, based on each of its following descriptions:

As a graph:

Solution

This fails the vertical line test, for example, at x = 0 a vertical line hits the graph at two points, which from the graph are approximately at y = 10 and y = 100.
As a list of ordered pairs.

Solution

There are two ordered pairs, (0, 13) and (0, 103) that have the same first value, but different second values. So there is not a single output for an input of 0.
As a formula: .

Solution

If we solve this for the output variable, y, by subtracting x² from, taking square roots of, and adding 58 to both sides, we are left with the equation . This is not a well-defined formula, in that it can take more than one y-value (output) for any given x-value (input).

As a table of values:

x	y
0 0 45 -45 27 27 · · ·	103 13 58 58 94 22 · · ·

Solution: The first and second rows (as well as the fifth and sixth rows) have the same value in the first column, but different values in the second. There is not a single output for several inputs.

As an arrow diagram. Note: While you will not be able to given the entire diagram (since that would have an infinite number of entries), you should draw enough to support your explanation.

Solution

From the following diagram:

We can see that there is more than one arrow coming from several elements on the left, such as 0 and 27, so it does not give a definite rule for taking inputs to outputs.

Back to Exercises.

Decide whether or not each of the following relations is a function, and explain your answer to your partner, using one of the tests from the text.
1. The relation given by the table:
  
  x y
  
  -3
  -2
  -1
  0
  1
  2
  3 2
  0.5
  0
  0.5
  2
  4.5
  8
  
  Solution
  
  There are no repeated values in the first column, so this does give a well-define rule for giving y-values for any given x-value; it is a function. Notice that a function may take the same output more than once (compare this with the example given in the text).
2. The relation which is the reverse of that in part a).
  
  Solution
  
  The reverse would be given by the table:
  
  x y
  
  2
  0.5
  0
  0.5
  2
  4.5
  8 -3
  -2
  -1
  0
  1
  2
  3
  
  The first and fifth rows (as well as the second and fourth rows) have the same value in the first column, but different values in the second. There is not a single output for several inputs, so this is not a function.
3. The relation given by the following arrow diagram:
4. The relation given by ordered pairs (x, y) which satisfy the equation x + 2y = 21.
  
  Solution
  
  Since we can solve for y (by subtracting x from both sides, then dividing both sides by 2) to obtain a single formula y = (21 - x)/2, which gives a single output (y-value) for each input (x-value). Thus, this is a functional relationship.
5. The relation given by the following arrow diagram:
  
  Solution
  
  This fails to express a function for two reasons, For some "inputs" (such as "a" and "b"), there are more than one "output" values. For others (i.e., "c"), there is not any "output" value defined. Thus, this fails to give exactly one "output" for every "input"".
Back to Exercises.
Decide whether or not each of the relations that you constructed in the exercises for the previous section, and explain your answer to your partner, using one of the tests from the text. Explain your answer to your partner, using one of the tests from the text.

Solution

Your partner should check your solutions.

Back to Exercises.

Go to Functional Notation and Terminology.

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Send questions or comments to jwicks@northpark.edu

Glossary

x	y
-3 -2 -1 0 1 2 3	2 0.5 0 0.5 2 4.5 8

x	y
2 0.5 0 0.5 2 4.5 8	-3 -2 -1 0 1 2 3