### All College Algebra Resources

## Example Questions

### Example Question #1 : Symmetry

Determine the symmetry of the following equation.

**Possible Answers:**

Symmetry along all axes.

Symmetry along the y-axis.

Symmetry along the origin.

Symmetry along the x-axis.

Does not have symmetry.

**Correct answer:**

Does not have symmetry.

To check for symmetry, we are going to do three tests, which involve substitution. First one will be to check symmetry along the x-axis, replace .

This isn't equivilant to the first equation, so it's not symmetric along the x-axis.

Next is to substitute .

This is not the same, so it is not symmetric along the y-axis.

For the last test we will substitute , and

This isn't the same as the orginal equation, so it is not symmetric along the origin.

The answer is it is not symmetric along any axis.

### Example Question #2 : Symmetry

Which of the following is true of the relation graphed above?

**Possible Answers:**

It is not a function

It is an odd function

It is an even function

It is a function, but it is neither even nor odd.

**Correct answer:**

It is an odd function

The relation graphed is a function, as it passes the vertical line test - no vertical line can pass through it more than once, as is demonstrated below:

Also, it can be seen to be symmetrical about the origin. Consequently, for each in the domain, - the function is odd.

### Example Question #3 : Symmetry

Which of the following is true of the relation graphed above?

**Possible Answers:**

It is an even function

It is not a function

It is an odd function

It is a function, but it is neither even nor odd.

**Correct answer:**

It is an odd function

The relation graphed is a function, as it passes the vertical line test - no vertical line can pass through it more than once, as is demonstrated below:

Also, it is seen to be symmetric about the origin. Consequently, for each in the domain, - the function is odd.

### Example Question #4 : Symmetry

is an even function; .

True or false: It follows that .

**Possible Answers:**

True

False

**Correct answer:**

False

A function is even if and only if, for all in its domain, . It follows that if , then

.

No restriction is placed on any other value as a result of this information, so the answer is false.

### Example Question #5 : Symmetry

The above table refers to a function with domain .

Is this function even, odd, or neither?

**Possible Answers:**

Even

Neither

Odd

**Correct answer:**

Neither

A function is odd if and only if, for every in its domain, ; it is even if and only if, for every in its domain, .

We see that and . Therefore, , so is false for at least one . cannot be even.

For a function to be odd, since , it follows that ; since is its own opposite, must be 0. However, ; cannot be odd.

The correct choice is neither.

### Example Question #6 : Symmetry

Define .

Is this function even, odd, or neither?

**Possible Answers:**

Neither

Odd

Even

**Correct answer:**

Neither

A function is odd if and only if, for all , ; it is even if and only if, for all , . Therefore, to answer this question, determine by substituting for , and compare it to both and .

, so is not even.

, so is not odd.

### Example Question #7 : Symmetry

is a piecewise-defined function. Its definition is partially given below:

How can be defined for negative values of so that is an odd function?

**Possible Answers:**

cannot be made odd.

**Correct answer:**

, by definition, is an odd function if, for all in its domain,

, or, equivalently

One implication of this is that for to be odd, it must hold that . If , then, since

for nonnegative values, then, by substitution,

This condition is satisfied.

Now, if is negative, is positive. it must hold that

,

so for all

,

the correct response.

### Example Question #7 : Symmetry

Consider the relation graphed above. Which is true of this relation?

**Possible Answers:**

The relation is an even function.

The relation not a function.

The relation is a function which is neither even nor odd.

The relation is an odd function.

**Correct answer:**

The relation is a function which is neither even nor odd.

The relation passes the Vertical Line test, as seen in the diagram below, in that no vertical line can be drawn that intersects the graph than once:

An function is odd if and only if its graph is symmetric about the origin, and even if and only if its graph is symmetric about the -axis. From the diagram, we see neither is the case.

### Example Question #8 : Symmetry

is a piecewise-defined function. Its definition is partially given below:

How can be defined for negative values of so that is an odd function?

**Possible Answers:**

**Correct answer:**

, by definition, is an odd function if, for all in its domain,

, or, equivalently

One implication of this is that for to be odd, it must hold that . Since is explicitly defined to be equal to 0 here, this condition is satisfied.

Now, if is negative, is positive. it must hold that

,

so for all

This is the correct choice.

### Example Question #9 : Symmetry

Which of the following is symmetrical to across the origin?

**Possible Answers:**

**Correct answer:**

Symmetry across the origin is symmetry across .

Determine the inverse of the function. Swap the x and y variables, and solve for y.

Subtract 3 on both sides.

Divide by negative two on both sides.

The answer is:

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