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1.4 Inverse Functions

3 min read•june 24, 2024

Inverse functions flip the roles of input and output, allowing us to "undo" a function's effect. They're crucial for solving equations and understanding relationships between variables. We use tools like the to check if a function is invertible.

help us find angles from ratios, with specific domains and ranges. When we combine a function with its inverse, we get the identity function. This relationship is key in calculus, especially when dealing with derivatives and solving complex equations.

Inverse Functions

Horizontal line test for invertibility

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  • Determines if a function is one-to-one meaning each x-value corresponds to exactly one y-value and vice versa
    • Functions that are not one-to-one fail the because some x-values map to multiple y-values (parabola)
  • Involves drawing horizontal lines across the graph of the function
    • If any horizontal line intersects the graph more than once, the function is not invertible ()
    • If no horizontal line intersects the graph more than once, the function is invertible ()
  • A function that passes the horizontal line test is injective

Computation and graphing of inverses

  • Steps to find the inverse of a function f(x)f(x):
    1. Replace f(x)f(x) with yy
    2. Interchange xx and yy variables
    3. Solve the resulting equation for yy
    4. Replace yy with f−1(x)f^{-1}(x) to denote the
  • The of f−1f^{-1} is the of ff and the range of f−1f^{-1} is the of ff
    • For example, if f(x)=x2f(x) = x^2 with domain [0,∞)[0, \infty), then f−1(x)=xf^{-1}(x) = \sqrt{x} with domain [0,∞)[0, \infty)
  • Graphing inverse functions involves reflecting the graph of ff across the line y=xy = x
    • If (2,4)(2, 4) is a point on the graph of ff, then (4,2)(4, 2) is a point on the graph of f−1f^{-1}

Applications of inverse trigonometric functions

  • allow you to find the angle given a trigonometric ratio
    • sin⁡−1(x)\sin^{-1}(x) or arcsin⁡(x)\arcsin(x) outputs the angle whose sine is xx
    • cos⁡−1(x)\cos^{-1}(x) or arccos⁡(x)\arccos(x) outputs the angle whose cosine is xx
    • tan⁡−1(x)\tan^{-1}(x) or arctan⁡(x)\arctan(x) outputs the angle whose tangent is xx
  • The domain of inverse trigonometric functions is limited to the range of their corresponding trigonometric function
    • sin⁡−1(x)\sin^{-1}(x) has domain [−1,1][-1, 1] and range [−π2,π2][-\frac{\pi}{2}, \frac{\pi}{2}] (first and fourth quadrants)
    • cos⁡−1(x)\cos^{-1}(x) has domain [−1,1][-1, 1] and range [0,π][0, \pi] (first and second quadrants)
    • tan⁡−1(x)\tan^{-1}(x) has domain (−∞,∞)(-\infty, \infty) and range (−π2,π2)(-\frac{\pi}{2}, \frac{\pi}{2}) (first and fourth quadrants)
  • Example calculation: If cos⁡(θ)=−32\cos(\theta) = -\frac{\sqrt{3}}{2} and θ\theta is in the third quadrant, then θ=cos⁡−1(−32)+π\theta = \cos^{-1}(-\frac{\sqrt{3}}{2}) + \pi

Functions vs their inverses

  • Composing a function with its inverse results in the identity function
    • (f∘f−1)(x)=x(f \circ f^{-1})(x) = x for all xx in the domain of f−1f^{-1} (plugging in f−1(x)f^{-1}(x) into f(x)f(x) returns xx)
    • (f−1∘f)(x)=x(f^{-1} \circ f)(x) = x for all xx in the domain of ff (plugging in f(x)f(x) into f−1(x)f^{-1}(x) returns xx)
  • The derivative of an is related to the derivative of the original function
    • If ff is differentiable and invertible, then f−1f^{-1} is also differentiable
    • (f−1)′(x)=1f′(f−1(x))(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))} (reciprocal of the derivative of ff evaluated at f−1(x)f^{-1}(x))

Equations with inverse functions

  • Inverse functions can be used to solve equations of the form f(x)=af(x) = a
    1. Apply f−1f^{-1} to both sides of the equation: f−1(f(x))=f−1(a)f^{-1}(f(x)) = f^{-1}(a)
    2. Simplify the left side using the property (f−1∘f)(x)=x(f^{-1} \circ f)(x) = x: x=f−1(a)x = f^{-1}(a)
  • Inverse trigonometric functions can be used to solve trigonometric equations
    • To solve cos⁡(x)=−22\cos(x) = -\frac{\sqrt{2}}{2}, apply cos⁡−1\cos^{-1} to both sides:
      1. cos⁡−1(cos⁡(x))=cos⁡−1(−22)\cos^{-1}(\cos(x)) = \cos^{-1}(-\frac{\sqrt{2}}{2})
      2. x=cos⁡−1(−22)+2πnx = \cos^{-1}(-\frac{\sqrt{2}}{2}) + 2\pi n or x=−cos⁡−1(−22)+2πnx = -\cos^{-1}(-\frac{\sqrt{2}}{2}) + 2\pi n, where nn is an integer

Function properties and invertibility

  • A function is surjective if every element in the codomain is mapped to by at least one element in the domain
  • A monotonic function is either entirely increasing or entirely decreasing, making it invertible
  • (f ∘ g) can be used to create new functions, which may or may not be invertible depending on the properties of f and g

Key Terms to Review (26)

Arccos: The inverse cosine function, denoted as arccos or cos^-1, is a trigonometric function that gives the angle whose cosine is equal to a given real number. It is the inverse of the cosine function, allowing one to find the angle given the cosine value.
Arcsin: Arcsin, also known as the inverse sine function, is a mathematical function that allows us to find the angle whose sine is a given value. It is the inverse of the sine function, which maps angles to their corresponding sine values. Arcsin is a crucial concept in the study of inverse functions and their derivatives.
Arctan: Arctan, also known as the inverse tangent function, is a mathematical function that represents the angle whose tangent is a given value. It is a fundamental inverse trigonometric function that is essential in understanding the behavior of inverse functions and their derivatives.
Bijective: A bijective function is a one-to-one correspondence between two sets, where each element in the domain is paired with a unique element in the codomain, and vice versa. This means that for every input, there is exactly one corresponding output, and for every output, there is exactly one corresponding input.
Composition: Composition is the act of combining or arranging multiple elements, functions, or operations into a unified whole. It is a fundamental concept in mathematics and various fields, describing how different components interact and integrate to form a cohesive structure or process.
Cubic Function: A cubic function is a polynomial function of degree three, where the highest exponent of the independent variable is three. Cubic functions are characterized by their S-shaped curve and can exhibit a variety of behaviors, including having one, two, or three real roots, depending on the function's coefficients.
Domain: The domain of a function is the set of all possible input values (typically $x$-values) for which the function is defined. It represents all the values that can be plugged into the function without causing any undefined behavior.
Domain: The domain of a function refers to the set of all possible input values for which the function is defined. It represents the range of values that the independent variable can take on, and it is a crucial concept in understanding the behavior and properties of functions.
Exponential Function: An exponential function is a mathematical function where the variable appears as the exponent. These functions exhibit a characteristic pattern of growth or decay, making them important in various fields of study, including calculus, physics, and finance.
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), returns the original input value. The inverse function is a way to 'undo' the original function, allowing us to solve for the input variable when given the output.
Function Composition: Function composition is the process of combining two or more functions to create a new function. The resulting function represents the combined effect of applying the individual functions in a specific order.
Horizontal line test: The horizontal line test is a method to determine if a function has an inverse that is also a function. A function passes the test if and only if no horizontal line intersects its graph more than once.
Horizontal Line Test: The horizontal line test is a graphical technique used to determine whether a function is one-to-one, or injective. It involves drawing horizontal lines across the graph of a function and observing the number of points where the line intersects the function's graph.
Inverse function: An inverse function is a function that reverses the effect of the original function. If $f(x)$ is a function, then 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 relationship between the input and output of another function. It takes the output of the original function and produces the corresponding input, effectively undoing the original function's operation.
Inverse trigonometric functions: Inverse trigonometric functions are the inverse operations of the trigonometric functions (sine, cosine, tangent, etc.), used to find the angle that corresponds to a given trigonometric value. These functions are commonly denoted as $\sin^{-1}(x)$, $\cos^{-1}(x)$, and $\tan^{-1}(x)$.
Inverse Trigonometric Functions: Inverse trigonometric functions are the inverse operations of the basic trigonometric functions (sine, cosine, tangent, etc.). They allow us to find the angle given the value of a trigonometric function, which is essential in various applications of calculus, such as finding derivatives and integrals involving trigonometric functions.
Invertibility: Invertibility is a fundamental property of functions that allows for the reversal of the input-output relationship. In the context of 1.4 Inverse Functions, invertibility is a crucial concept that enables the construction of inverse functions, which are essential for solving a variety of mathematical problems.
Logarithmic function: A logarithmic function is the inverse of an exponential function and is typically written as $y = \log_b(x)$, where $b$ is the base. It represents the power to which the base must be raised to obtain a given number.
Logarithmic Function: A logarithmic function is a mathematical function that describes an exponential relationship between two quantities. It is the inverse of an exponential function, allowing for the representation of quantities that grow or decay at a constant rate over time. Logarithmic functions are essential in various fields, including mathematics, science, and engineering, for their ability to model and analyze complex phenomena.
Natural exponential function: The natural exponential function is defined as $e^x$, where $e$ is Euler's number, approximately equal to 2.71828. It is a fundamental function in calculus with unique properties related to growth and decay.
One-to-one function: A one-to-one function, also known as an injective function, is a type of function in which every element of the domain maps to a unique element in the codomain. This implies that no two different inputs can map to the same output.
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 range. In other words, for any two distinct elements in the domain, their corresponding elements in the range must also be distinct.
Range: Range refers to the set of all possible output values (or 'y' values) that a function can produce based on its domain (the set of input values). Understanding the range helps us grasp how a function behaves, what outputs are attainable, and the limitations on those outputs.
Reflection: Reflection is a mathematical transformation that involves mirroring a function or graph across a line or axis. It is a fundamental concept that applies to various topics in calculus, including the study of basic function classes and inverse functions.
Restricted domain: A restricted domain is a subset of the original domain of a function, chosen to make the function one-to-one. This is essential for finding an inverse function.
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Glossary
Glossary
1.5 Exponential and Logarithmic Functions