Chapter 6: Problem 1

Explain what is meant by the inverse of a function.

Short Answer

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Answer: The inverse of a function is another function that essentially reverses or undoes the actions of the original function. It is denoted as f^{-1}. For a function to have an inverse, it must be both injective (one-to-one) and surjective (onto), making it a bijective function.

Step by step solution

Definition of the Inverse of a Function

In mathematics, the inverse of a function is another function that essentially "reverses" or "undoes" the actions of the original function. In a more formal definition, for a function f, its inverse function, denoted as f^{-1}, is a function such that for every input-output pair (x, y) in f, there is a corresponding input-output pair (y, x) in f^{-1}. In other words, if f(x)=y, then f^{-1}(y)=x.

Example of an Inverse Function

Let's consider the function f(x) = 2x + 3. To find the inverse of this function, we first need to solve for x in terms of y. So, let y = 2x + 3. Solving for x, we get x = (y - 3)/2. Therefore, the inverse function f^{-1}(y) = (y - 3)/2.

Graphical Representation

If you were to graph both a function and its inverse on the same set of axes, you'd find that the two graphs are reflections of each other across the line y=x. This is because the x and y-values have switched places between the function and its inverse.

Conditions for a Function to Have an Inverse

For a function to have an inverse, it must be both injective (one-to-one) and surjective (onto). Injective means that every input has a unique output, while surjective means that every output has a corresponding input. If a function is both injective and surjective, it is called a bijective function. Bijective functions have unique inverses.

The Relationship Between a Function and Its Inverse

A function and its inverse have a special property: when the inverse function is applied to the output of the original function, the original input is obtained. Mathematically speaking, this means that if f(x)=y, then f^{-1}(f(x)) = f^{-1}(y) = x. Additionally, if f^{-1}(y) = x, then f(f^{-1}(y)) = f(x) = y. These properties hold true for all valid inputs and outputs in the domain and range of the function and its inverse.

Notation for the Inverse of a Function

The notation for the inverse of a function is typically written as f^{-1}(x) (note that this does NOT mean 1/f(x)). Keep in mind that this is different from the reciprocal of a function, which would be written as 1/f(x) or f(x)^{-1}.

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Explain what is meant by the inverse of a function.

Short Answer

Step by step solution

Definition of the Inverse of a Function

Example of an Inverse Function

Graphical Representation

Conditions for a Function to Have an Inverse

The Relationship Between a Function and Its Inverse

Notation for the Inverse of a Function

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