3.5 Inverse functions

Inverse functions are mathematical tools that undo the effects of other functions. They play a crucial role in complex analysis, allowing us to reverse operations and solve equations. Understanding inverse functions is essential for grasping more advanced concepts in the field.

This topic covers the definition, existence, and properties of inverse functions. We'll explore methods for finding inverses, their graphical representations, and important theorems. We'll also examine examples and applications, highlighting their significance in mathematics and real-world problem-solving.

Definition of inverse functions

• Inverse functions are functions that "undo" each other
• If $f(x)$ and $g(x)$ are inverse functions, then $f(g(x)) = g(f(x)) = x$ for all $x$ in the domain of the respective functions
• The notation for the inverse of a function $f(x)$ is $f^{-1}(x)$, read as "f inverse of x"

Existence and uniqueness

One-to-one functions

Top images from around the web for One-to-one functions

• 

  Identify a One-to-One Function | Intermediate Algebra View original

  Is this image relevant?

• 

  What is an inverse function? - Mathematics for Teaching View original

  Is this image relevant?

• 

  Inverse and Composite Functions | Boundless Algebra View original

  Is this image relevant?

• 

  Identify a One-to-One Function | Intermediate Algebra View original

  Is this image relevant?

• 

  What is an inverse function? - Mathematics for Teaching View original

  Is this image relevant?

1 of 3

Top images from around the web for One-to-one functions

• 

  Identify a One-to-One Function | Intermediate Algebra View original

  Is this image relevant?

• 

  What is an inverse function? - Mathematics for Teaching View original

  Is this image relevant?

• 

  Inverse and Composite Functions | Boundless Algebra View original

  Is this image relevant?

• 

  Identify a One-to-One Function | Intermediate Algebra View original

  Is this image relevant?

• 

  What is an inverse function? - Mathematics for Teaching View original

  Is this image relevant?

1 of 3

• A function is considered one-to-one (or injective) if each element in the codomain is paired with at most one element in the domain
• In other words, no two distinct elements in the domain map to the same element in the codomain
• One-to-one functions are necessary for the existence of an inverse function

Horizontal line test

• The horizontal line test is a graphical method to determine if a function is one-to-one
• If any horizontal line intersects the graph of a function more than once, the function is not one-to-one and, therefore, does not have an inverse
• Conversely, if no horizontal line intersects the graph more than once, the function is one-to-one and has an inverse

Finding inverse functions

Algebraic method

• To find the inverse of a function algebraically, follow these steps:
  1. Replace $f(x)$ with $y$
  2. Swap $x$ and $y$
  3. Solve the equation for $y$
  4. Replace $y$ with $f^{-1}(x)$
• Example: Given $f(x) = 3x - 2$, find $f^{-1}(x)$
  1. $y = 3x - 2$
  2. $x = 3y - 2$
  3. $x + 2 = 3y$, then $y = \frac{x + 2}{3}$
  4. $f^{-1}(x) = \frac{x + 2}{3}$

Graphical method

• The graph of an inverse function is obtained by reflecting the graph of the original function over the line $y = x$
• This reflection is equivalent to swapping the $x$ and $y$ coordinates of each point on the graph
• Example: The graph of $f(x) = x^2$ and its inverse $f^{-1}(x) = \sqrt{x}$ are symmetric about the line $y = x$

Properties of inverse functions

Domain and range

• The domain of a function becomes the range of its inverse, and the range of a function becomes the domain of its inverse
• This is because the inverse function "undoes" the original function, mapping the output values back to their corresponding input values

Symmetry about the line y = x

• The graphs of a function and its inverse are symmetric about the line $y = x$
• This symmetry is a result of swapping the $x$ and $y$ coordinates when finding the inverse function
• If a point $(a, b)$ lies on the graph of $f(x)$, then the point $(b, a)$ lies on the graph of $f^{-1}(x)$

Composition of functions and inverses

Inverses as "undoing" functions

• The composition of a function and its inverse, in either order, results in the identity function
• If $f(x)$ and $g(x)$ are inverse functions, then $f(g(x)) = g(f(x)) = x$ for all $x$ in the domain of the respective functions
• This property demonstrates how inverse functions "undo" each other, returning the input value when composed

Continuity of inverse functions

• If a function $f(x)$ is continuous and one-to-one on an interval, then its inverse $f^{-1}(x)$ is also continuous on the corresponding interval
• The continuity of the inverse function is a consequence of the continuity and one-to-one property of the original function
• Example: The natural logarithm function $\ln(x)$ is continuous and one-to-one on the interval $(0, \infty)$, so its inverse, the exponential function $e^x$, is also continuous on the real numbers

Differentiation of inverse functions

Derivative of an inverse function

• If a function $f(x)$ is differentiable and one-to-one on an interval, then its inverse $f^{-1}(x)$ is also differentiable on the corresponding interval
• The derivative of the inverse function is given by: $\frac{d}{dx}f^{-1}(x) = \frac{1}{f'(f^{-1}(x))}$
• In other words, the derivative of the inverse function is the reciprocal of the derivative of the original function, evaluated at the inverse function

Inverse function theorem

• The inverse function theorem states that if a function $f(x)$ is differentiable and has a non-zero derivative at a point $a$, then its inverse $f^{-1}(x)$ is also differentiable at the point $f(a)$
• This theorem provides a way to find the derivative of an inverse function without explicitly finding the inverse function itself
• Example: Given $f(x) = e^x$, find the derivative of $f^{-1}(x)$ at $x = 0$
  • $f'(x) = e^x$
  • $f^{-1}(x) = \ln(x)$
  • By the inverse function theorem, $\frac{d}{dx}\ln(x)|_{x=1} = \frac{1}{f'(f^{-1}(1))} = \frac{1}{e^{\ln(1)}} = 1$

Examples of inverse functions

Exponential and logarithmic functions

• The exponential function $f(x) = a^x$ (where $a > 0$ and $a \neq 1$) and the logarithmic function $g(x) = \log_a(x)$ are inverse functions
• For example, the natural exponential function $f(x) = e^x$ and the natural logarithm function $g(x) = \ln(x)$ are inverses of each other
• These functions are continuous and differentiable on their respective domains

Trigonometric and inverse trigonometric functions

• The trigonometric functions (sine, cosine, tangent) have inverse functions called inverse trigonometric functions (arcsine, arccosine, arctangent)
• For example, the sine function $f(x) = \sin(x)$ and the arcsine function $g(x) = \arcsin(x)$ are inverses of each other
• These functions are continuous and differentiable on restricted domains to ensure one-to-one correspondence

Applications of inverse functions

Solving equations

• Inverse functions can be used to solve equations by applying the inverse function to both sides of the equation
• For example, to solve the equation $e^x = 5$, apply the natural logarithm (the inverse of the exponential function) to both sides: $\ln(e^x) = \ln(5)$ $x = \ln(5)$
• This technique is particularly useful when dealing with exponential, logarithmic, or trigonometric equations

Modeling real-world phenomena

• Inverse functions are used to model various real-world phenomena, particularly in cases where there is a need to "undo" a process
• For example, in finance, the concept of present value (PV) and future value (FV) are related by inverse functions:
  • Given an interest rate $r$ and a number of periods $n$, the future value is calculated as $FV = PV(1 + r)^n$
  • To find the present value, the inverse function is used: $PV = \frac{FV}{(1 + r)^n}$
• Other applications include modeling population growth, radioactive decay, and many other natural processes