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Example Questions
Example Question #1 :年代ymmetry
Determine the symmetry of the following equation.
年代ymmetry along the y-axis.
年代ymmetry along the x-axis.
年代ymmetry along the origin.
年代ymmetry along all axes.
Does not have symmetry.
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 #61 :Graphs
Which of the following is true of the relation graphed above?
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.
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 eachin the domain,- the function is odd.
Example Question #1 :年代ymmetry
Which of the following is true of the relation graphed above?
It is not a function
It is an even function
It is a function, but it is neither even nor odd.
It is an odd function
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 eachin the domain,- the function is odd.
Example Question #63 :Graphs
is an even function;.
True or false: It follows that.
False
True
False
A functionis even if and only if, for allin 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 :年代ymmetry
The above table refers to a functionwith domain.
Is this function even, odd, or neither?
Even
Neither
Odd
Neither
A functionis odd if and only if, for everyin its domain,; it is even if and only if, for everyin its domain,.
We see thatand. Therefore,, sois false for at least one.cannot be even.
For a function to be odd, since, it follows that; sinceis its own opposite,must be 0. However,;cannot be odd.
The correct choice is neither.
Example Question #1 :年代ymmetry
Define.
Is this function even, odd, or neither?
Neither
Odd
Even
Neither
A functionis odd if and only if, for all,; it is even if and only if, for all,. Therefore, to answer this question, determineby substitutingfor, and compare it to bothand.
, sois not even.
, sois not odd.
Example Question #7 :年代ymmetry
is a piecewise-defined function. Its definition is partially given below:
How canbe defined for negative values ofso thatis an odd function?
cannot be made odd.
, by definition, is an odd function if, for allin its domain,
, or, equivalently
One implication of this is that forto be odd, it must hold that. If, then, since
for nonnegative values, then, by substitution,
This condition is satisfied.
Now, ifis negative,is positive. it must hold that
,
so for all
,
the correct response.
Example Question #1 :年代ymmetry
Consider the relation graphed above. Which is true of this relation?
The relation is an odd function.
The relation not a function.
The relation is a function which is neither even nor odd.
The relation is an even function.
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 #2 :年代ymmetry
is a piecewise-defined function. Its definition is partially given below:
How canbe defined for negative values ofso thatis an odd function?
, by definition, is an odd function if, for allin its domain,
, or, equivalently
One implication of this is that forto be odd, it must hold that. Sinceis explicitly defined to be equal to 0 here, this condition is satisfied.
Now, ifis negative,is positive. it must hold that
,
so for all
This is the correct choice.
Example Question #1 :年代ymmetry
Which of the following is symmetrical toacross the origin?
年代ymmetry across the origin is symmetry across.
Determine the inverse of the function. Swap the x and y variables, and solve for y.
年代ubtract 3 on both sides.
Divide by negative two on both sides.
The answer is:
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