Symmetry - College Algebra
Card 0 of 12
Determine the symmetry of the following equation.
Determine the symmetry of the following equation.
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.
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
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.
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Which of the following is true of the relation graphed above?
Which of the following is true of the relation graphed above?
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.
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
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The above table refers to a function 
with domain 
.
Is this function even, odd, or neither?
The above table refers to a function
Is this function even, odd, or 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.
A function
We see that
For a function to be odd, since
The correct choice is neither.
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Which of the following is true of the relation graphed above?
Which of the following is true of the relation graphed above?
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.
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
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Define 
.
Is this function even, odd, or neither?
Define
Is this function even, odd, or 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.
A function
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is an even function; 
.
True or false: It follows that 
.
True or false: It follows that
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.
A function
No restriction is placed on any other value as a result of this information, so the answer is false.
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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?
How can
, 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.
One implication of this is that for
Now, if
so for all
This is the correct choice.
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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?
How can
, 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.
One implication of this is that for
for nonnegative values, then, by substitution,
This condition is satisfied.
Now, if
so for all
the correct response.
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Consider the relation graphed above. Which is true of this relation?
Consider the relation graphed above. Which is true of this relation?
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.
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
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Which of the following is symmetrical to 
across the origin?
Which of the following is symmetrical to
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: 
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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Consider the function 
.
Is 
an even function, an odd function, or neither?
Consider the function
Is
A function 
is even if, for each 
in its domain,
.
It is odd if, for each 
in its domain,
.
Substitute 
for 
in the definition:
Since 
, 
is an even function.
A function
It is odd if, for each
Substitute
Since
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is an even function. Let 
.
Is 
an even function, an odd function, or neither?
Is
A function is even if, for each in its domain,
.
It is odd if, for each in its domain,
.
Substitute 
for 
in the definition of 
:
Since is even, 
, so
This makes 
an odd function.
A function
It is odd if, for each
Substitute
Since
This makes
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