Is $lnleft( frac{1+2x}{1-2x}right)$ an odd or even function?
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Is $$f:left(-frac{1}{2},frac{1}{2}right) to mathbb{R}, x mapsto lnleft( frac{1+2x}{1-2x}right)$$ an odd or even function?
The function can be decomposed this way:
$$f(x)=ln(1+2x)-ln(1-2x)$$
![[Graph of the function]](https://i.stack.imgur.com/5qm2r.png)
It seems that this function is an odd function but I dont know how to justify it.
real-analysis functions logarithms even-and-odd-functions
edited Nov 18 at 9:37
Viktor Glombik
489321
asked Nov 18 at 9:27
user605734 MBS
1319
add a comment |
up vote
1
down vote
favorite
Is $$f:left(-frac{1}{2},frac{1}{2}right) to mathbb{R}, x mapsto lnleft( frac{1+2x}{1-2x}right)$$ an odd or even function?
The function can be decomposed this way:
$$f(x)=ln(1+2x)-ln(1-2x)$$
It seems that this function is an odd function but I dont know how to justify it.
real-analysis functions logarithms even-and-odd-functions
edited Nov 18 at 9:37
Viktor Glombik
489321
asked Nov 18 at 9:27
user605734 MBS
1319
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Is $$f:left(-frac{1}{2},frac{1}{2}right) to mathbb{R}, x mapsto lnleft( frac{1+2x}{1-2x}right)$$ an odd or even function?
The function can be decomposed this way:
$$f(x)=ln(1+2x)-ln(1-2x)$$
It seems that this function is an odd function but I dont know how to justify it.
real-analysis functions logarithms even-and-odd-functions
edited Nov 18 at 9:37
Viktor Glombik
489321
asked Nov 18 at 9:27
user605734 MBS
1319
Is $$f:left(-frac{1}{2},frac{1}{2}right) to mathbb{R}, x mapsto lnleft( frac{1+2x}{1-2x}right)$$ an odd or even function?
The function can be decomposed this way:
$$f(x)=ln(1+2x)-ln(1-2x)$$
It seems that this function is an odd function but I dont know how to justify it.
real-analysis functions logarithms even-and-odd-functions
real-analysis functions logarithms even-and-odd-functions
edited Nov 18 at 9:37
Viktor Glombik
489321
asked Nov 18 at 9:27
user605734 MBS
1319
edited Nov 18 at 9:37
Viktor Glombik
489321
asked Nov 18 at 9:27
user605734 MBS
1319
edited Nov 18 at 9:37
Viktor Glombik
489321
edited Nov 18 at 9:37
Viktor Glombik
489321
edited Nov 18 at 9:37
Viktor Glombik
489321
489321
asked Nov 18 at 9:27
user605734 MBS
1319
asked Nov 18 at 9:27
user605734 MBS
1319
asked Nov 18 at 9:27
user605734 MBS
1319
1319
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
A odd function must satisfy the equation $f(-x) = -f(x)$.
Let $x in left(-frac{1}{2}, frac{1}{2}right)$.
Then we have, as you rightly observed
begin{align*}
-f(x)
= ln(1 - 2x) - ln(1 + 2x)
= ln(1 + 2(-x)) - ln(1 - 2(-x))
= f(-x).
end{align*}
Note that a function being odd graphically corresponds to rotational symmetry. Similarly to the graph of $x mapsto x^3$ (the standard example in my opinion), when you rotate the graph 180° with respect to the origin (alternatively flip over the $y$- and then the $x$-axis), you end up with the same graph.
answered Nov 18 at 9:29
Viktor Glombik
489321
so the function doesn't need to be defined in all $mathbb{R}$ to be an odd or even function?
– user605734 MBS
Nov 18 at 9:39
That's correct. You could check out the wikipedia definition, where the above mentioned property has to be valid for all $x$ in the domain of $f$.
– Viktor Glombik
Nov 18 at 9:41
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
A odd function must satisfy the equation $f(-x) = -f(x)$.
Let $x in left(-frac{1}{2}, frac{1}{2}right)$.
Then we have, as you rightly observed
begin{align*}
-f(x)
= ln(1 - 2x) - ln(1 + 2x)
= ln(1 + 2(-x)) - ln(1 - 2(-x))
= f(-x).
end{align*}
Note that a function being odd graphically corresponds to rotational symmetry. Similarly to the graph of $x mapsto x^3$ (the standard example in my opinion), when you rotate the graph 180° with respect to the origin (alternatively flip over the $y$- and then the $x$-axis), you end up with the same graph.
answered Nov 18 at 9:29
Viktor Glombik
489321
so the function doesn't need to be defined in all $mathbb{R}$ to be an odd or even function?
– user605734 MBS
Nov 18 at 9:39
That's correct. You could check out the wikipedia definition, where the above mentioned property has to be valid for all $x$ in the domain of $f$.
– Viktor Glombik
Nov 18 at 9:41
add a comment |
up vote
1
down vote
accepted
A odd function must satisfy the equation $f(-x) = -f(x)$.
Let $x in left(-frac{1}{2}, frac{1}{2}right)$.
Then we have, as you rightly observed
begin{align*}
-f(x)
= ln(1 - 2x) - ln(1 + 2x)
= ln(1 + 2(-x)) - ln(1 - 2(-x))
= f(-x).
end{align*}
Note that a function being odd graphically corresponds to rotational symmetry. Similarly to the graph of $x mapsto x^3$ (the standard example in my opinion), when you rotate the graph 180° with respect to the origin (alternatively flip over the $y$- and then the $x$-axis), you end up with the same graph.
answered Nov 18 at 9:29
Viktor Glombik
489321
so the function doesn't need to be defined in all $mathbb{R}$ to be an odd or even function?
– user605734 MBS
Nov 18 at 9:39
That's correct. You could check out the wikipedia definition, where the above mentioned property has to be valid for all $x$ in the domain of $f$.
– Viktor Glombik
Nov 18 at 9:41
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
A odd function must satisfy the equation $f(-x) = -f(x)$.
Let $x in left(-frac{1}{2}, frac{1}{2}right)$.
Then we have, as you rightly observed
begin{align*}
-f(x)
= ln(1 - 2x) - ln(1 + 2x)
= ln(1 + 2(-x)) - ln(1 - 2(-x))
= f(-x).
end{align*}
Note that a function being odd graphically corresponds to rotational symmetry. Similarly to the graph of $x mapsto x^3$ (the standard example in my opinion), when you rotate the graph 180° with respect to the origin (alternatively flip over the $y$- and then the $x$-axis), you end up with the same graph.
answered Nov 18 at 9:29
Viktor Glombik
489321
A odd function must satisfy the equation $f(-x) = -f(x)$.
Let $x in left(-frac{1}{2}, frac{1}{2}right)$.
Then we have, as you rightly observed
begin{align*}
-f(x)
= ln(1 - 2x) - ln(1 + 2x)
= ln(1 + 2(-x)) - ln(1 - 2(-x))
= f(-x).
end{align*}
Note that a function being odd graphically corresponds to rotational symmetry. Similarly to the graph of $x mapsto x^3$ (the standard example in my opinion), when you rotate the graph 180° with respect to the origin (alternatively flip over the $y$- and then the $x$-axis), you end up with the same graph.
answered Nov 18 at 9:29
Viktor Glombik
489321
edited Nov 18 at 9:45
answered Nov 18 at 9:29
Viktor Glombik
489321
answered Nov 18 at 9:29
Viktor Glombik
489321
answered Nov 18 at 9:29
Viktor Glombik
489321
489321
so the function doesn't need to be defined in all $mathbb{R}$ to be an odd or even function?
– user605734 MBS
Nov 18 at 9:39
That's correct. You could check out the wikipedia definition, where the above mentioned property has to be valid for all $x$ in the domain of $f$.
– Viktor Glombik
Nov 18 at 9:41
add a comment |
so the function doesn't need to be defined in all $mathbb{R}$ to be an odd or even function?
– user605734 MBS
Nov 18 at 9:39
That's correct. You could check out the wikipedia definition, where the above mentioned property has to be valid for all $x$ in the domain of $f$.
– Viktor Glombik
Nov 18 at 9:41
so the function doesn't need to be defined in all $mathbb{R}$ to be an odd or even function?
– user605734 MBS
Nov 18 at 9:39
so the function doesn't need to be defined in all $mathbb{R}$ to be an odd or even function?
– user605734 MBS
Nov 18 at 9:39
That's correct. You could check out the wikipedia definition, where the above mentioned property has to be valid for all $x$ in the domain of $f$.
– Viktor Glombik
Nov 18 at 9:41
That's correct. You could check out the wikipedia definition, where the above mentioned property has to be valid for all $x$ in the domain of $f$.
– Viktor Glombik
Nov 18 at 9:41
add a comment |
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