For example, if f(x) = x2 for all x (that is, f is the squaring function) and g(x) = x + 1 (that is, g is the linear function with rate of change and initial value both 1), then both f ∘ g and g ∘ f are linear coordinate transformations of f.
In particular, (f ∘ g)(x) = (x + 1)2; this is called a passive or inside coordinate transformation. On the other hand, (g ∘ f)(x) = x2 + 1; this is called an active or outside coordinate transformation.
More generally, instead of a non-constant linear function, we could use any invertible function with a sufficiently large domain or range. (A non-constant linear function is always invertible, and its domain and range always consist of all real numbers.)
More concretely, consider these examples:
|Coordinate transformation of f:||Effect on the graph:|
|f(x) + 1,||Shift 1 unit upwards;|
|f(x) − 1,||Shift 1 unit downwards;|
|2f(x),||Stretch vertically by a factor of 2;|
|f(x)/2,||Compress vertically by a factor of 2;|
|−f(x),||Flip vertically across the horizontal axis;|
|−2f(x),||Flip and stretch vertically;|
|2f(x) + 1,||Stretch vertically and then shift upwards;|
|f(x + 1),||Shift 1 unit to the left;|
|f(x − 1),||Shift 1 unit to the right;|
|f(2x),||Compress horizontally by a factor of 2;|
|f(x/2),||Stretch horizontally by a factor of 2;|
|f(−x),||Flip horizontally across the vertical axis;|
|f(−2x),||Flip and compress horizontally;|
|f(2x + 1),||Shift to the left and then compress horizontally;|
|2f(x + 1),||Stretch vertically and shift to the left, in either order.|
This web page was written between 2010 and 2015 by Toby Bartels, last edited on 2015 October 30. Toby reserves no legal rights to it.
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