1 | initial version |

Let me denote `2*(x^3-2)`

by `f`

and `sign(0.5*x+2.0)*abs(0.5*x+2.0)^(1/3)`

by `g`

.

Since `f`

and `g`

are inverse to each other, there is a symmetry of the whole graphs with respect to the line {x=y}, but you do not see it.

First, you need to have the same scale for x and y axes, you can get this by setting the option `aspect_ratio=1`

.

Second, you look at the interval [-4,0] for both functions while this interval is not invariant: `f([-4,0])`

is not equal to `[-4,0]`

, so the graph of `g`

on `[-4,0]`

has no reason to be symmetric of the graph of `f`

on `[-4,0]`

. If you want to see the symmetry, you need to plot the graph of `g`

on the image of `[-4,0]`

by `f`

.

```
sage: f = 2*(x^3-2)
sage: g = sign(0.5*x+2.0)*abs(0.5*x+2.0)^(1/3)
sage: f(-4)
-132
sage: f(0)
-4
sage: F = plot ((2*(x^3-2)),-4,0, aspect_ratio=1)
sage: G = plot(sign(0.5*x+2.0)*abs(0.5*x+2.0)^(1/3),-132,-4, aspect_ratio=1)
sage: F+G
```

So, you can see that `f:[-4,0] -> [-132,-4]`

is a bijection whose inverse is `g: [-132,-4] -> [-4,0]`

.

The plot is even nicer for the map `f:[-3,3] -> [-58,50]`

:

```
sage: f(-3)
-58
sage: f(3)
50
sage: F = plot ((2*(x^3-2)),-3,3, aspect_ratio=1)
sage: G = plot(sign(0.5*x+2.0)*abs(0.5*x+2.0)^(1/3),-58,50, aspect_ratio=1)
sage: F+G
```

Conclusion : maps are not only formulas, but require domain and codomain to be well defined !

2 | No.2 Revision |

Let me denote `2*(x^3-2)`

by `f`

and `sign(0.5*x+2.0)*abs(0.5*x+2.0)^(1/3)`

by `g`

.

Since `f`

and `g`

are inverse to each ~~other, ~~other (on the whole real line), there is a symmetry of the whole graphs with respect to the line {x=y}, but you do not see it.

First, you need to have the same scale for x and y axes, you can get this by setting the option `aspect_ratio=1`

.

Second, you look at the interval [-4,0] for both functions while this interval is not invariant: `f([-4,0])`

is not equal to `[-4,0]`

, so the graph of `g`

on `[-4,0]`

has no reason to be symmetric of the graph of `f`

on `[-4,0]`

. If you want to see the symmetry, you need to plot the graph of `g`

on the image of `[-4,0]`

by `f`

.

```
sage: f = 2*(x^3-2)
sage: g = sign(0.5*x+2.0)*abs(0.5*x+2.0)^(1/3)
sage: f(-4)
-132
sage: f(0)
-4
sage: F = plot ((2*(x^3-2)),-4,0, aspect_ratio=1)
sage: G = plot(sign(0.5*x+2.0)*abs(0.5*x+2.0)^(1/3),-132,-4, aspect_ratio=1)
sage: F+G
```

So, you can see that `f:[-4,0] -> [-132,-4]`

is a bijection whose inverse is `g: [-132,-4] -> [-4,0]`

.

The plot is even nicer for the map `f:[-3,3] -> [-58,50]`

:

```
sage: f(-3)
-58
sage: f(3)
50
sage: F = plot ((2*(x^3-2)),-3,3, aspect_ratio=1)
sage: G = plot(sign(0.5*x+2.0)*abs(0.5*x+2.0)^(1/3),-58,50, aspect_ratio=1)
sage: F+G
```

Conclusion : maps are not only formulas, but require domain and codomain to be well defined !

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