- Subject:
- Mathematics
- Material Type:
- Module
- Provider:
- Ohio Open Ed Collaborative
- Tags:
- License:
- Creative Commons Attribution Non-Commercial
- Language:
- English
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Chapter 5.2 - Inverse Functions
Khan Academy - Inverse Functions
Paul's Online Notes - Inverse Functions
Inverse Functions
Inverse of a Function
Emphasize the relationship between the domain and range of a one-to-one function and its inverse. Make sure that students understand the difference between y and x as notated in y = f(x) and y = f-1(x).
Finding inverses of quadratic and rational functions is a good exercise.
See https://blogs.ams.org/matheducation/2016/11/28/inverse-functions-were-teaching-it-all-wrong/
Recall that a function between two variables, x and y, describes the pattern of dependence of one variable, say y, in terms of the other variable x. This pattern tells us how x is transformed to become y. An inverse function, if it exists, inverts this pattern and tells us how y must be transformed to recover x. For example,
- Subtraction is the inverse of addition and vice-versa
- Division is the inverse of multiplication and vice-versa
- Square-root is the inverse of squaring and vice-versa
- Etc.
If y = f(x) is a function of x, then we know that to each given x value there is exactly one y value. This is verified graphically by the vertical line test. For the existence of the inverse function of y = f(x), it must be also true that for each given y there must be exactly one x value. This property is verified by the horizontal line test. If a function f passes both vertical and horizontal line tests then f is called a one-to-one function. It is a fact that inverse functions exist for one-to-one functions. For example, the linear function y = mx + b, m ≠ 0, is a one-to-one function and its inverse is x = (y – b)/m.
Not all functions are one-to-one. For example, y = f(x) = x2, -∞ < x < ∞, is not one-to-one since the value y is same for both x and -x. However, by suitable restriction of the domain a function can be made into a one-to-one function. For example, restricting x to the set of non-negative numbers, y = f(x) = x2 becomes one-to-one. The inverse of this function is
\(x = f^{-1}(y) = \sqrt{y}\),
where y ≥ 0.
Review: Composition of functions, function arithmetic, completing-the-square method for quadratic functions, solving rational equations.
Learning Objectives: In this module we will learn
- One-to-one function: verify if a function f is one-to-one both graphically and algebraically.
- Determine the domain and range of a one-to-one function f and its inverse f-1.
- Algebraically determine the inverse of a given one-to-one function.
- Algebraically verify the properties of an inverse function
- Determine the graph of inverse function f-1 from the graph of f.
- Apply inverse functions.