The document defines inverse functions and provides examples. An inverse function f-1(x) undoes the original function f(x) so that f-1(f(x)) = x. For a function to have an inverse, it must be one-to-one meaning each output of f(x) corresponds to only one input x. The document gives examples of linear functions that are invertible and the function y=x2 that is not invertible because it is not one-to-one. It also states that if a function f(x) is one-to-one on its domain, then it has an inverse function and the domain of f(x) is equal to the range of the
The Sum of Two Functions
The Difference of Two functions
The Product of Two Functions
The Quotient of Two Functions
The Product of A constant and a Function
The Sum of Two Functions
The Difference of Two functions
The Product of Two Functions
The Quotient of Two Functions
The Product of A constant and a Function
A PowerPoint presentation on the Derivative as a Function. Includes example problems on finding the derivative using the definition, Power Rule, examining graphs of f(x) and f'(x), and local linearity.
Properties of Functions
Odd and Even Functions
Periodic Functions
Monotonic Functions
Bounded Functions
Maxima and Minima of Functions
Inverse Function
Sequence and Series
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
2. Definition
Let f (x) have domain D and range R.
The inverse function f -1 (x) (if it exists)
is the function with domain R such that
f -1 (f (x)) = x for x ϵ D
f (f -1 (x)) = x for x ϵ R
If f -1 exists then f is called invertible.
3. Example: Linear Function
Let f (x) = 4x – 1. Find f -1 (x) and show
that
f (x) is invertible.
f -1 (x) = ¼ (x + 1)
x f (x) x f -1 (x)
0 -1 -1 0
2 7 7 2
-2 -9 -9 -2
3 11 11 3
171 683 683 171
6. Example: Function with no
Inverse
y = x2
Is y = √x the inverse??
Is y = ±√x the inverse?
x y = x2 x y = √x x y = ±√x
-2 4 4 2 4 ±2
-1 1 1 1 1 ±1
0 0 0 0 0 0
1 1 1 1 1 ±1
2 4 4 2 4 ±2
8. One-to-one Function
Definition: A function f (x) is one-to-
one (on its domain D) if for every
value c, the equation f (x)=c has at
most one solution for x ϵ D.
a c
Domain of f = Range of f -1 Range of f = Domain of f -1
9. Theorem 1: Existence of
Inverses
If f (x) is one-to-one on its domain D
then f is invertible.
Domain of f = range of f -1
Range of f = domain of f -1
10. Derivative of the Inverse
Assume f (x) is invertible and one-to-
one with the inverse
g (x) = f -1 (x). If b belongs to the
domain of g (x) and f ’ (g (b)) ≠ 0 then
g ’ (b) exists and: