Inverse of Functions DL  M4, S2, A3 
DL  M4, S3, A1 
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Note: inverse means opposite

Interactive Activity

Example
Graph f(x) = x^{2} + 3 and its inverse.Solution
Replace f(x) with y for ease in finding the inverse.
y = x^{2} + 3
Find the inverse.
Note: The graphs of a an original function and it's inverse are always reflections in the line y = x.
Notice from the graph that the relation x = y^{2} + 3 is not a function, since y is not uniquely determined for x > 3. In other words there are more than one yvalue for each xvalue in the region x > 3.
If you were to restrict the original function to y = x^{2} + 3, where x > 0, then its inverse would be x = y^{2} + 3, where y > 0. The negative portions of each graph would be eliminated. As a result, the inverse could be rewritten as or .
Recall that the inverse will undo what the function does. For example, the function takes 4 into 19; that is, f(4) = 19. The inverse will take 19 back into 4; that is, f^{ 1}(19) = 4.
There are many ways of restricting the domain of a function so that its inverse is also a function. In the previous example, you could have written f(x) = x^{2} + 3, x < 0. The inverse would be x = y^{2} + 3 + 3, y < 0. When you solve for y, the inverse is or .
Other Examples:


The domain of f(x) is the set of reals,and the range is f(x) > 1. The domain of f^{ 1}(x) is x > 1and the range is the set of reals,.



Inverse Check
Parts of this work has been adapted from a Math 20 Pure learning resource originally produced and owned by Alberta Education