Inverse Functions: One-to-One Functions and Their Applications

4.1 Inverse Functions

One-to-One Functions

Understanding one-to-one functions is essential for determining whether a function has an inverse. A function is one-to-one if each output value is paired with only one input value.

• Definition: A function f is one-to-one if for every a and b in the domain, f(a) = f(b) implies a = b.

• Horizontal Line Test (HLT): A function is one-to-one if every horizontal line intersects its graph at most once.

Example: Determine whether each function is one-to-one:

• $f(x) = rac{1}{x-12}$

• $f(x) = \sqrt{x-7}$

Use the horizontal line test on their graphs to decide.

Inverse Functions

An inverse function reverses the effect of the original function. If f is a one-to-one function, its inverse is denoted f^{-1}.

• Definition: If f is one-to-one, then its inverse f^{-1} satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

• Notation: f^{-1}(x) indicates the inverse function of f.

• Domain and Range: The domain of f is the range of f^{-1}, and the range of f is the domain of f^{-1}.

Example: Determine whether $f(x) = e^{x-1}$ and $g(x) = \ln(x) + 1$ are inverses.

Example: Find the inverse of each function that is one-to-one:

• $F = \{(-1,1), (0,2), (1,3), (2,5)\}$

• $G = \{(1,2), (2,1), (2,2), (1,0)\}$

Finding Equations of Inverses

To find the equation of an inverse function, follow these steps:

1. Switch x and y in the equation.

2. Solve for y.

3. Replace y with f^{-1}(x).

Example: Determine whether each equation defines a one-to-one function. If so, find the equation of the inverse.

• (a) $f(x) = 2x + 5$

• (b) $f(x) = 1x - 2^x$

• (c) $g = x^4 + 2$

Example: The following rational function is one-to-one. Find its inverse:

• $f(x) = \frac{2x+3}{x-4}, \ x \neq 4$

Graphing Inverse Functions

The graph of an inverse function is a reflection of the original function's graph across the line $y = x$.

• Given the graph of a function, graph its inverse by reflecting each point over the line $y = x$.

Example: Let $f(x) = \sqrt{x-5}, \ x \geq 5$. Find $f^{-1}(x)$ and graph both functions.

Applications of Inverse Functions

Inverse functions are used in real-world applications such as decoding messages or solving equations for unknowns.

• Example: The function $f(x) = 2x - 9$ was used to encode a message as a sequence of numbers. To decode, find the inverse function and apply it to each number.

Letter	Encoded Value
A	11
B	13
C	15
D	17
E	19
F	21
G	23
H	25
I	27
J	29
K	31
L	33
M	35
N	37
O	39
P	41

To decode, use the inverse function $f^{-1}(x) = \frac{x+9}{2}$.

Additional info: The notes cover the concept of one-to-one functions, the definition and properties of inverse functions, methods for finding inverses, graphical interpretation, and a practical application involving encoding and decoding messages.