Inverse Functions

Inverse of a Function

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.
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