College Algebra : Symmetry

Example Questions

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Example Question #1 : Symmetry

Determine the symmetry of the following equation.

Symmetry along the y-axis.

Does not have symmetry.

Symmetry along the origin.

Symmetry along all axes.

Symmetry along the x-axis.

Does not have symmetry.

Explanation:

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?

It is an even function

It is an odd function

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

It is not a function

It is an odd function

Explanation:

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?

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

It is an even function

It is not a function

It is an odd function

It is an odd function

Explanation:

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 .

True

False

False

Explanation:

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 #2 : Symmetry

The above table refers to a function  with domain .

Is this function even, odd, or neither?

Even

Neither

Odd

Neither

Explanation:

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 #3 : Symmetry

Define .

Is this function even, odd, or neither?

Even

Neither

Odd

Neither

Explanation:

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?

Explanation:

, 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 #4 : Symmetry

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

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

The relation is an odd function.

The relation not a function.

The relation is an even function.

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

Explanation:

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 #9 : 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?

Explanation:

, 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 #5 : Symmetry

Which of the following is symmetrical to  across the origin?

Explanation:

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.