Inverse Functions
If f is a function, the inverse function f−1 (if it
exists) undoes whatever f does. That is,
y=f−1(x)⟺x=f(y)
and
x=f−1(y)⟺y=f(x) |
A few facts:
- f(f−1(◻))=◻, no matter what you put into the box.
- The domain of f−1 is the range of f; and the range of
f−1 is the domain of f.
So, the independent variable of f is now the dependent
variable of f−1 and vice versa. Graphically, this
means that the horizontal axis becomes vertical, and the
vertical axis horizontal.
- The function f−1 exists only if f is one-to-one.
Graphically, f−1 is a function if its graph passes
the vertical line test, which means that the graph of f must
pass the horizontal line test.
If f is not one-to-one, we must restrict the domain of f
first.
For instance, if f(x)=x+1, then the inverse function is
f−1(x)=x−1, since subtracting 1 undoes the effect of adding
1. If f(x)=2x, then f−1(x)=x/2, since dividing by 2
undoes the effect of multiplying by 2. We'll explore how to
find the formula for f−1(x) in general on the next slide.
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