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Lecture 1 : Inverse functions
One-to-one Functions A function f is one-to-one if it never takes the same value twice or
f (x 1 ) 6 = f (x 2 ) whenever x 1 6 = x 2.
Example The function f (x) = x is one to one, because if x 1 6 = x 2 , then f (x 1 ) 6 = f (x 2 ).
On the other hand the function g(x) = x^2 is not a one-to-one function, because g(โ1) = g(1).
Graph of a one-to-one function If f is a one to one function then no two points (x 1 , y 1 ), (x 2 , y 2 ) have the same y-value. Therefore no horizontal line cuts the graph of the equation y = f (x) more than once.
Example Compare the graphs of the above functions
Determining if a function is one-to-one
Horizontal Line test: A graph passes the Horizontal line test if each horizontal line cuts the graph at most once.
Using the graph to determine if f is one-to-one A function f is one-to-one if and only if the graph y = f (x) passes the Horizontal Line Test.
Example Which of the following functions are one-to-one?
Using the derivative to determine if f is one-to-one A continuous function whose derivative is always positive or always negative is a one-to-one function. Why?
Example Is the function g(x) =
4 x + 4 a one-to-one function?
Inverse functions
Inverse Functions If f is a one-to-one function with domain A and range B, we can define an inverse function f โ^1 (with domain B ) by the rule
f โ^1 (y) = x if and only if f (x) = y.
This is a sound definition of a function, precisely because each value of y in the domain of f โ^1 has exactly one x in A associated to it by the rule y = f (x).
Example If f (x) = x^3 + 1, use the equivalence of equations given above find f โ^1 (9) and f โ^1 (28).
Note that the domain of f โ^1 equals the range of f and the range of f โ^1 equals the domain of f.
Example Let g(x) =
4 x + 4. What is Domain g? What is Range g?
Does gโ^1 exist?
What is Domain gโ^1?
What is Range gโ^1?
What is gโ^1 (4)?
Finding a Formula For f โ^1 (x)
Given a formula for f (x), we would like to find a formula for f โ^1 (x). Using the equivalence
x = f โ^1 (y) if and only if y = f (x)
we can sometimes find a formula for f โ^1 using the following method:
- In the equation y = f (x), if possible solve for x in terms of y to get a formula x = f โ^1 (y).
- Switch the roles of x and y to get a formula for f โ^1 of the form y = f โ^1 (x).
Example Let f (x) = (^2) xxโ+1 3 , find a formula for f โ^1 (x).
We can derive properties of the graph of y = f โ^1 (x) from properties of the graph of y = f (x), since they are refections of each other in the line y = x. For example:
Theorem If f is a one-to-one continuous function defined on an interval, then its inverse f โ^1 is also one-to-one and continuous. (Thus f โ^1 (x) has an inverse, which has to be f (x), by the equivalence of equations given in the definition of the inverse function.)
Theorem If f is a one-to-one differentiable function with inverse function f โ^1 and f โฒ(f โ^1 (a)) 6 = 0, then the inverse function is differentiable at a and
(f โ^1 )โฒ(a) =
f โฒ(f โ^1 (a))
proof y = f โ^1 (x) if and only if x = f (y). Using implicit differentiation we differentiate x = f (y) with respect to x to get
1 = f โฒ(y)
dy dx
or
f โฒ(y)
dy dx
or
f โฒ(y)
= (f โ^1 )โฒ(x) or
f โฒ(f โ^1 (x))
= (f โ^1 )โฒ(x)
Geometrically this means that if (a, f โ^1 (a)) is a point on the curve y = f โ^1 (x), then the point (f โ^1 (a), a) is on the curve y = f (x) and the slope of the tangent to the curve y = f โ^1 (x) at (a, f โ^1 (a)) is the reciprocal of the tangent to the curve y = f (x) at the point (f โ^1 (a), a). The graphs of the function f (x) = (^2) xxโ+1 3 and f โ^1 (x) = (^3) xxโ+1 2 are shown below. You can verify that โ7 = (f โ^1 )โฒ(3) = (^) f โฒ(10)^1.
Note To use the above formula for (f โ^1 )โฒ(a), you do not need the formula for f โ^1 (x), you only need the value of f โ^1 at a and the value of f at f โ^1 (a).
Example Consider the function f (x) =
4 x + 4 defined above. Find (f โ^1 )โฒ(4).
What does the formula from the theorem say?
Use the equivalence of the equations y = f โ^1 (x) and x = f (y) to find f โ^1 (4).
Put this in the formula from the theorem to find (f โ^1 )โฒ(4).
Example Let f (x) = x^3 + 1, find (f โ^1 )โฒ(28).
Example If f is a one-to-one function with the following properties:
f (10) = 21, f โฒ(10) = 2, f โ^1 (10) = 4. 5 , f โฒ(4.5) = 3.
Find (f โ^1 )โฒ(10).
