Derivatives of Inverse Functions

INVERSE FUNCTIONS

In some cases, the effects of a function can be "undone" by another function.

Example. Let f be the linear function defined by f(x)=3x + 1. Given a real number x, the function f first multiplies x by three and then adds one. This process is reversed by the linear function g(x)=(x-1)/3 that subtracts one and divides by three.

g(f(x)) = g(3x+1) = ((3x+1)-1)/3 = x
The same thing happens in the other order; given a real number x, the composite function f o g simply returns x.

f(g(x)) = f((x-1)/3) = 3(x-1)/3 + 1 = x
If we begin with x=2, then we find that f(2)=7. Applying g to 7 brings us right back to where we started: g(7)=2. These two calculations show up on the graphs of these functions in this way:

The point (2,7) is on the graph of f.
The point (7,2) is on the graph of g.

In overall graphical terms, the graphs of f and g are related by a reflection across the line y=x. This is a standard feature of inverse functions.

**Definition of Inverse Functions**
The functions f and g are inverse functions if the following two statements are true. g(f(x)) = x for all x in the domain of f f(g(x)) = x for all x in the domain of g

Notation. When f and g are inverse functions, we write g = f^-1 and f = g^-1
Thus the essential point is that f^-1(f(x))=x for all x in the domain of f and that f(f^-1(x))=x for all x in the domain of f^-1. One notes that (f^-1)^-1 = f.

Example (revisited). If f(x)=3x+1 then f^-1(x)=(x-1)/3.

WARNING: The inverse function f^-1 and the reciprocal function 1/f are very different objects. The inverse of the linear function f(x)=3x+1 is a linear function; the reciprocal function is nonlinear:y=1/(3x+1). The graph of the reciprocal has a horizontal asymptote at the x-axis and a vertical asymptote at x=-1/3.

General Features of Inverse Functions. Suppose that f has an inverse function f^-1.

f(x) = y if and only if f^-1(y) =x.
The point (x,y) is on the graph of f if and only if the point (y,x) is on the graph of f^-1.
The graph of f is obtained from the graph of f^-1 by reflection across the line y=x...and vice versa.
The domain of f is equal to the image of f^-1. The domain of f^-1 is equal to the image of f.

Existence of Inverse Functions. Not every function f has an inverse f^-1. For example, if we take f to be the squaring function f(x)=x², we find that the points (3,9) and (-3,9) are both on the graph of f. (This is because f(3)=9=f(-3).) If there were an inverse function f^-1, then the points (9,3) and (9,-3) would both have to be on the graph of f^-1. This impossible by the vertical line test: no vertical line can touch the graph of a function in more than one point.

If we reflect a vertical line across y=x we obtain a horizontal line. A consequence of this is the following horizontal line test for the existence of an inverse function.

**Horizontal Line Test**
The function f has an inverse function f^-1 if and only if each horizontal line in the x-y plane touches the graph of f at most once.

If the graph of f passes the horizontal line test (HLT), then when we reflect the graph across the line y=x, we obtain a curve that passes the vertical line test (VLT). This reflected curve is then the graph of the inverse function f^-1.

We now consider some standard examples of inverse functions. In many cases, we construct inverse functions by restricting the domain of a given function in order to force its graph to pass the HLT.

Root Functions

Let n be a positive integer and let f be the power function of exponent n: f(x)=xⁿ. We can restrict the domain of f so that the graph of f passes the HLT. The inverse function is called the nth root function.

f^-1(x) =

= x^1/n
For display purposes on the Web, it is convenient for us to use the fractional exponent notation. If n=2, this is the square root function. For n=3 this is the cube root function.

If n is even, then the graph of f(x)=xⁿ does not pass the horizontal line test, and so it is necessary to restrict the domain of f in this case to the interval [0,). If n is even, then domain of the nth root function is the interval [0,). The domain of the nth root function is the entire real line if n is odd.

**y = sqrt(x) = x^1/2**

**y = x^1/3**

Inverse Trigonometric Functions

By restricting the domain of the sine function to values of x between -/2 and /2, we obtain a graph that passes the horizontal line test. The resulting inverse function to the sine function is called the arcsine function or the inverse sine function.

y = arcsin(x) = sin^-1(x)
The domain of the arcsine function is the closed interval [-1,1]. The graph of the restricted sine function and the arcsine function are shown below.

y = sin(x)
-/2 x /2

y = arcsin(x)
-1 x 1

By restricting the domain of the tangent function y=tan(x)=sin(x)/cos(x) to values of x strictly between -

/2 and

/2, we obtain a graph that passes the horizontal line test. The resulting inverse function to the tangent function is called the arctangent function or the inverse tangent function.

y = arctan(x) = tan^-1(x)
The domain of the arctangent function is the entire real line. The graph of the restricted tangent function and the arctangent function are shown below. Notice how the vertical asymptotes in the graph of the restricted tangent function become distinct horizontal asymptotes for the graph of the arctangent function.

y = tan(x)
-/2 < x < /2

y = arctan(x)
- < x <

We now consider the problem of differentiating inverse functions.

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