Inverse Functions
Introduction
One-to-one functions, or invertible functions, are functions that have inverses. One-to-one functions have a nice test to tell from the graph if a function is one-to-one. The definition of a one-to-one function is given through the equation of the function. The conceptual idea for a function being one-to-one is that for every number in the range there is one number in the domain that the function maps to the point in the range.
Definition
A function is one-to-one if two distinct values in the domain get mapped to the same value in the range. More concretely, if f is one-to-one and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1 \text{ and }x_2} are two numbers in the domain of f such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x_1) = f(x_2)} then we must have that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1 = x_2}
Examples: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x, x^3, 3x + 2, }
Non-examples: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2, \left| x \right|}
Horizontal Line Test
It was mentioned earlier that there is a way to tell if a function is one-to-one from its graph. This method is called the horizontal line test. It is the same as the vertical line test, except we use a horizontal line. So a function is one-to-one if every horizontal line crosses the graph at most once.
Inverse Functions
If a function is one-to-one, then to each y in the range of f there is a unique x in the domain that maps on top of it. So we can define a function from the range of f back to the domain. This new function is called the inverse function, and is denoted Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}} .
Properties: Let f be a one-to-one function, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}} be its inverse. a) Domain of f = Range of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}} b) Range of f = Domain of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}} c) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(f(x)) = x} where x is in the domain of f^{-1} d) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(f^{-1}(x)) = x } where x is in the domain of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(x)}
Since we have both a graphical and algebraic method to determine if a function is one-to-one, we also have geometric and algebraic methods to find the inverse.
The graphical method is to reflect the graph of f across the line y = x. Although easy to explain, the process is harder to visualize. The algebraic method is the opposite, harder to explain but easier to execute.
The algebraic method can be explained in 3 steps:
Step 1) In the equation y = f(x), interchange all instances of x with the variable y, and vice versa.
Step 2) If possible, solve for y.
Step 3) Replace y with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(x)} and check that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(f(x)) = x \text{ and }f(f^{-1}(x)) = x}
Example: Find the inverse of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \frac{1}{x + 3}}
We start by swapping the x and y to get Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = \frac{1}{y + 3}}
Now we can solve for y and swap y for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(x)} . Doing so we will find that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(x) = \frac{1-3x}{x}}