• Use to determine whether a function is
• A function is one-to-one if and only if no
horizontal line intersects its graph more
Graph f(x) = −3x + 4.
the graph at the
function is one-to-
one and thus
has an inverse
that is a function.
graph at the
and thus has
an inverse that
is a function.
How do you find an inverse?
• “Undo” the function.
• Replace the x with y and solve for y .
How to find the Inverse of a
1. Replace f(x) with y in the equation.
2. Interchange x and y in the equation.
3. Solve thisequation for y .
4. Replace y with f-1(x). Any restrictions on x or y should be considered and
included with the equation.
Remember: Domain and Range are interchanged
Determine whether the function f(x) = 3x − 2
is one-to-one, and if it is, find a formula for
• If a function is one-to-one, then its
inverse is a function.
• The domain of a one-to-one function f is
the range of the inverse f-1.
• The range of a one-to-one function f is
the domain of the inverse f-1.
• A function that is increasing over its
domain or is decreasing over its domain
is a one-to-one function.
Restricting a Domain
• When the inverse of a function is not a
function, the domain of the function can
be restricted to allow the inverse to be a
• In such cases, it is convenient to consider
“part” of the function by restricting the
domain of f(x). If the domain is restricted,
then its inverse is a function.
Restricting the Domain
Recall that if a function is not one-to-one,
then its inverse will not be a function.
If we restrict the domain values of f(x) to those
than or equal to zero , we see that f(x) is now one-to-one
and its inverse is now a function.
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