Inverse functions flip input and output, letting us undo what a function does. They're like mathematical opposites, reversing each other's effects. Understanding inverses helps us solve equations and model real-world situations where we need to work backwards.
We can find inverses algebraically by swapping x and y, or graphically by flipping over y=x. Domains and ranges swap too. Not all functions have inverses, so we sometimes need to restrict domains to make them work.
Inverse Functions
Verification of inverse functions
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Definition states two functions f and g are inverses if and only if:
Composing f(g(x)) equals x for all x values in the of g
Composing g(f(x)) equals x for all x values in the domain of f
Verify inverse functions by composing f(g(x)) and g(f(x))
If both compositions simplify to x, the functions are confirmed inverses
Example: Given f(x)=2x+1 and g(x)=2x−1, verify they are inverses
Evaluate inverse functions by finding the x value that makes f(x)=a
Example: If f(x)=x2 and f−1(9)=3, then f(3)=9
(e.g., f(x) and f−1(x)) is used to represent a function and its inverse
Domains and ranges of inverses
For a f with domain A and B:
The f−1 has domain B and range A
Domain and range are swapped for inverse functions
Some functions require domain restrictions to have an inverse
Restrict domain to make the function one-to-one
Square root functions: Restrict domain to x≥0
Logarithmic functions: Restrict domain to x>0
Example: f(x)=x2 is not one-to-one, but f(x)=x2 with x≥0 is one-to-one and has an inverse
Finding inverse functions
Algebraically find inverse functions:
Start with [y = f(x)](https://www.fiveableKeyTerm:y_=_f(x))
Swap x and y to get x=f(y)
Solve the equation for y in terms of x
Replace y with f−1(x) to get the inverse function
Example: Find the inverse of f(x)=3x−2
Graphically find inverse functions by reflecting the graph of f(x) over the line y=x
The reflected graph is the graph of f−1(x)
Example: The inverse of f(x)=2x is f−1(x)=log2x, found by reflecting f(x) over y=x
Graphing inverses by reflection
The line y=x acts as a line for the graphs of inverse functions
If (a,b) is a point on f(x), then (b,a) is a point on f−1(x)
Coordinates are swapped when reflecting over y=x
To graph inverse functions:
Graph the original function f(x)
Reflect each point of f(x) over the line y=x
The reflected points make up the graph of f−1(x)
Example: The graphs of f(x)=2x and f−1(x)=2x are reflections over y=x
This reflection demonstrates the between a function and its inverse
Properties of Inverse Functions
: A function must be both injective (one-to-one) and surjective (onto) to have an inverse
: Strictly increasing or decreasing functions always have inverses
(e.g., arcsin, arccos, arctan) are examples of functions with restricted domains to ensure
Key Terms to Review (21)
Bijection: A bijection is a one-to-one and onto function, meaning that for every element in the codomain, there is exactly one corresponding element in the domain, and every element in the domain is mapped to a unique element in the codomain. Bijections are a special type of function that establish a perfect correspondence between two sets.
Bijectivity: Bijectivity is a property of a function that describes a one-to-one correspondence between the elements of the domain and the elements of the codomain. In other words, each element in the domain is uniquely paired with one and only one element in the codomain, and vice versa.
Composition: Composition refers to the act of combining or putting together multiple elements or functions to create a new, unified whole. It is a fundamental concept that underpins various mathematical and logical operations, allowing for the exploration of relationships and the generation of new insights.
Domain: The domain of a function refers to the set of all possible input values for the function. It represents the range of values that the independent variable can take on. The domain is a crucial concept in understanding the behavior and properties of various mathematical functions.
Exponential Function: An exponential function is a mathematical function where the independent variable appears as the exponent. These functions exhibit a characteristic growth or decay pattern, with the value of the function increasing or decreasing at a rate that is proportional to the current value. Exponential functions are fundamental in understanding various real-world phenomena, from population growth to radioactive decay.
F^(-1)(x): The inverse function of a function f(x) is denoted as f^(-1)(x). It represents the function that, when applied to the output of f(x), results in the original input value. The inverse function allows you to 'undo' the original function, reversing the transformation and recovering the original input.
Function Notation: Function notation is a way of representing functions using symbolic expressions, where the function name is followed by an input value enclosed in parentheses. It provides a concise and efficient way to denote the relationship between the input and output of a function, allowing for the evaluation and manipulation of functions.
Horizontal Line Test: The horizontal line test is a method used to determine whether a function is one-to-one, or invertible. It involves drawing horizontal lines across the graph of a function to see if each horizontal line intersects the graph at no more than one point.
Inverse Function: An inverse function is a function that undoes the operation of another function. It is a special type of function that reverses the relationship between the input and output variables of the original function, allowing you to solve for the input when given the output.
Inverse Trigonometric Functions: Inverse trigonometric functions are the inverse operations of the standard trigonometric functions, allowing us to determine the angle given the ratio of the sides of a right triangle. They are essential in understanding and solving various trigonometric equations and problems.
Invertibility: Invertibility is a fundamental property of functions that describes the ability to reverse the relationship between the input and output values. In other words, it refers to the existence of an inverse function that can undo the original function's operation, restoring the original input from the given output.
Logarithmic Function: A logarithmic function is a function that describes an exponential relationship between two quantities, where one quantity is the logarithm of the other. It is the inverse of an exponential function and has applications in various fields, including mathematics, science, and engineering.
Monotonicity: Monotonicity is a property of a function that describes its behavior as the input variable increases or decreases. A function is considered monotonic if it either consistently increases or consistently decreases over its domain.
One-to-One Function: A one-to-one function, also known as an injective function, is a special type of function where each element in the domain is paired with a unique element in the codomain. In other words, for any two distinct elements in the domain, their corresponding elements in the codomain must also be distinct.
Range: The range of a function refers to the set of all possible output values or the set of all values that the function can attain. It represents the vertical extent or the interval of values that the function can produce as the input variable changes. The range is an important concept in the study of functions and their properties, as it provides information about the behavior and characteristics of the function.
Reflection: Reflection is a mathematical transformation that flips or mirrors a function or graph about a line, either the x-axis or the y-axis. This concept is essential in understanding the behavior and properties of various functions and their graphs.
Square Root Function: The square root function is a mathematical function that takes a non-negative real number and returns its positive square root. It is denoted by the symbol $\sqrt{}$ or $x^{1/2}$, where $x$ is the input value. The square root function is a fundamental concept in mathematics and has applications in various fields, including geometry, physics, and engineering.
Symmetry: Symmetry is a fundamental concept in mathematics and physics that describes the presence of regularity, balance, and pattern within an object or function. It refers to the ability of an object or function to be transformed in a certain way without altering its overall appearance or properties.
Vertical Line Test: The vertical line test is a graphical method used to determine whether a relation or function is a function. It involves drawing a vertical line through the graph and checking if the line intersects the graph at only one point for each value of the independent variable.
Y = f(x): The expression 'y = f(x)' represents a function, where 'y' is the dependent variable and 'x' is the independent variable. This notation indicates that the value of 'y' is determined by the value of 'x' through the function 'f'. The function 'f' is a rule or a relationship that maps each input value of 'x' to a unique output value of 'y'.
Y = x line: The y = x line, also known as the identity line or the line of equality, is a straight line that passes through the origin and has a slope of 1. It represents the set of points where the x-coordinate is equal to the y-coordinate, meaning that for any point on the line, y = x.