Glossary

3.7 Inverse Functions

2 min read•june 24, 2024

Inverse functions are mathematical partners that undo each other's operations. They swap domains and ranges, creating a mirror-like relationship. Understanding inverse functions helps us grasp and the reversibility of mathematical operations.

Calculating inverse functions involves flipping x and y, then solving for y. Graphing inverses is like reflecting the original function across y=x. These concepts are crucial for understanding function relationships and solving complex equations in algebra.

Inverse Functions

Inverse function determination

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  • Two functions f(x)f(x) and g(x)g(x) are inverses if they "undo" each other
    • Composing f(x)f(x) and g(x)g(x) in either order results in the original input (xx)
    • f(g(x))=xf(g(x)) = x for all xx in the of g(x)g(x) (undoes g(x)g(x))
    • g(f(x))=xg(f(x)) = x for all xx in the of f(x)f(x) (undoes f(x)f(x))
  • Checking if functions are inverses involves
    • Compose f(g(x))f(g(x)) and g(f(x))g(f(x)) algebraically or by substitution
    • If both compositions simplify to xx, the functions are inverses (f(x)=x2f(x) = x^2 and g(x)=xg(x) = \sqrt{x})
  • This process demonstrates function composition and the between functions

Domain and range of inverses

  • The domain and of inverse functions are swapped
    • Domain of f(x)f(x) becomes the range of f−1(x)f^{-1}(x) (f(x)=x2f(x) = x^2, domain: R\mathbb{R}, range: [0,∞)[0, \infty))
    • Range of f(x)f(x) becomes the domain of f−1(x)f^{-1}(x) (f−1(x)=xf^{-1}(x) = \sqrt{x}, domain: [0,∞)[0, \infty), range: R\mathbb{R})
  • One-to-one functions have inverses
    • Each input (xx) maps to a unique output (yy) ()
    • determines if a function is one-to-one
      • If any horizontal line intersects the graph more than once, the function is not one-to-one and has no inverse (f(x)=x2f(x) = x^2)

Calculation of inverse functions

  • Finding the inverse of a function f(x)f(x) involves
    1. Replace f(x)f(x) with yy
    2. Interchange xx and yy variables
    3. Solve the equation for yy
    4. Replace yy with f−1(x)f^{-1}(x)
  • Example: Inverse of f(x)=3x−4f(x) = 3x - 4
    1. y=3x−4y = 3x - 4
    2. x=3y−4x = 3y - 4
    3. x+4=3yx + 4 = 3y, x+43=y\frac{x + 4}{3} = y
    4. f−1(x)=x+43f^{-1}(x) = \frac{x + 4}{3}

Graphing inverses from originals

  • Graphing an involves reflecting the original function's graph
    • Reflect the graph of f(x)f(x) across the line y=xy = x to obtain the graph of f−1(x)f^{-1}(x)
    • The line y=xy = x serves as a symmetry line between a function and its inverse
  • process
    1. Choose points on the original function's graph (f(x)=x2f(x) = x^2, points: (0,0)(0, 0), (1,1)(1, 1), (2,4)(2, 4))
    2. Swap the xx and yy coordinates of each point ((0,0)(0, 0), (1,1)(1, 1), (4,2)(4, 2))
    3. Plot the new points and connect them to graph the (f−1(x)=xf^{-1}(x) = \sqrt{x})
  • property: If (a,b)(a, b) is on the graph of f(x)f(x), then (b,a)(b, a) is on the graph of f−1(x)f^{-1}(x) ((2,4)(2, 4) on f(x)f(x), (4,2)(4, 2) on f−1(x)f^{-1}(x))
  • This visually represents the inverse relationship between functions

Key Terms to Review (32)

Algebraic Manipulation: Algebraic manipulation refers to the process of performing various operations and transformations on algebraic expressions to simplify, solve, or rearrange them. It involves the strategic use of mathematical rules and properties to manipulate expressions and equations in a logical and systematic manner.
Bijective: A bijective function is a one-to-one correspondence between two sets, where each element in the domain is paired with exactly one element in the codomain, and vice versa. This type of function is also known as a one-to-one and onto function, as it establishes a unique relationship between the elements of the two sets.
Celsius: Celsius is a scale for measuring temperature where 0 degrees represents the freezing point of water and 100 degrees represents the boiling point of water under standard atmospheric conditions. It is used in many scientific contexts due to its precision and ease of use.
Composition: Composition refers to the way in which elements or parts are combined or arranged to form a whole. It is a fundamental concept in mathematics, particularly in the context of functions and their relationships.
Composition of functions: The composition of functions is the application of one function to the results of another. It is denoted as $(f \circ g)(x) = f(g(x))$.
Domain: The domain of a function is the complete set of possible input values (x-values) that allow the function to work within its constraints. It specifies the range of x-values for which the function is defined.
Domain: The domain of a function refers to the set of input values for which the function is defined. It represents the range of values that the independent variable can take on, and it is the set of all possible values that can be plugged into the function to produce a meaningful output.
Exponential function: An exponential function is a mathematical expression in the form $f(x) = a \cdot b^x$, where $a$ is a constant, $b$ is the base greater than 0 and not equal to 1, and $x$ is the exponent. These functions model growth or decay processes.
Exponential Function: An exponential function is a mathematical function in which the independent variable appears as an exponent. These functions exhibit a characteristic curve that grows or decays at a rate proportional to the current value, leading to rapid changes in output as the input increases.
F^(-1)(x): The inverse function of a function f(x) is denoted as f^(-1)(x). It represents the function that undoes the operation performed by the original function f(x), allowing the input to be recovered from the output.
Fahrenheit: Fahrenheit is a temperature scale where water freezes at 32 degrees and boils at 212 degrees under standard atmospheric conditions. It is often used in the United States for everyday temperature measurements.
Function Composition: Function composition is the process of combining two or more functions to create a new function. The output of one function becomes the input of the next function, allowing for the creation of more complex mathematical relationships.
Graphical Symmetry: Graphical symmetry refers to the visual symmetry observed in the graph of a function. It describes the various ways in which a function's graph can exhibit symmetrical properties, which is an important consideration when analyzing and understanding the behavior of a function.
Horizontal line test: The horizontal line test is a method used to determine if a function is one-to-one (injective). If any horizontal line intersects the graph of the function at most once, then the function passes the test and is one-to-one.
Horizontal Line Test: The horizontal line test is a method used to determine whether a function is one-to-one, meaning that each output value is associated with only one input value. It involves drawing horizontal lines across the graph of a function to see if the line intersects the graph at more than one point.
Horizontal reflection: A horizontal reflection is a transformation that flips a function's graph over the y-axis. It changes the sign of the x-coordinates of all points on the graph.
Injective: Injectivity is a property of a function where each element in the codomain (output) is mapped to by at most one element in the domain (input). In other words, an injective function is a one-to-one correspondence between the domain and the codomain.
Inverse function: An inverse function reverses the operation of a given function. If $f(x)$ is a function, its inverse $f^{-1}(x)$ satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.
Inverse Function: An inverse function is a function that reverses the operation of another function. It undoes the original function, mapping the output back to the original input. Inverse functions are crucial in understanding the relationships between different mathematical concepts, such as domain and range, composition of functions, transformations, and exponential and logarithmic functions.
Inverse Relationship: An inverse relationship is a relationship between two variables where an increase in one variable corresponds to a decrease in the other variable, and vice versa. This type of relationship is characterized by a negative correlation, where the variables move in opposite directions.
Invertible: Invertibility is a property of a function where the function has an inverse function. This means the function can be 'undone' or reversed, allowing the input and output values to be swapped. Invertibility is a crucial concept in the study of inverse functions and solving systems of equations using inverse matrices.
Invertible matrix: An invertible matrix is a square matrix that has an inverse, meaning there exists another matrix which, when multiplied with the original, yields the identity matrix. A matrix is invertible if and only if its determinant is non-zero.
Line y = x: The line y = x is a straight line that passes through the origin and forms a 45-degree angle with the positive x-axis. This line represents the set of points where the x-coordinate and y-coordinate are equal, meaning that for any point on the line, the value of y is the same as the value of x.
Logarithmic Function: A logarithmic function is a special type of function where the input variable is an exponent. It is the inverse of an exponential function, allowing for the determination of the exponent when the result is known. Logarithmic functions play a crucial role in various mathematical concepts and applications.
One-to-one function: A one-to-one function (injective function) is a function where each element of the domain maps to a unique element in the codomain. No two different elements in the domain map to the same element in the codomain.
One-to-One Function: A one-to-one function, also known as an injective function, is a function where each element in the domain is mapped to a unique element in the codomain. This means that for every input value, there is only one corresponding output value, and no two input values can be mapped to the same output value.
Range: In mathematics, the range refers to the set of all possible output values (dependent variable values) that a function can produce based on its input values (independent variable values). Understanding the range helps in analyzing how a function behaves and what values it can take, connecting it to various concepts like transformations, compositions, and types of functions.
Reflection: Reflection is a transformation of a function that creates a mirror image of the original function across a specified axis. This concept is fundamental in understanding the behavior and properties of various mathematical functions.
Restricting the domain: Restricting the domain involves limiting the set of input values (x-values) for which a function is defined to ensure it meets certain criteria, such as being one-to-one. This is often necessary when finding the inverse of a function.
Symmetric: Symmetric refers to a property where an object or function exhibits a balance or regularity in its form or arrangement. In the context of mathematics, symmetry is a fundamental concept that describes the invariance of an object or function under certain transformations, such as reflection, rotation, or translation.
Uniqueness: Uniqueness refers to the quality of being one of a kind, distinct, or unparalleled. In the context of inverse functions, uniqueness describes the property where each element in the domain of a function corresponds to a single, distinct element in the range of the function.
Y = f^(-1)(x): The expression 'y = f^(-1)(x)' represents the inverse function of the original function 'f(x)'. The inverse function is a transformation that 'undoes' the original function, allowing you to find the input value given the output value.
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