college algebra 5.4 inverses of functions

Section 5.4: Inverses of Functions

Objectives

• Understanding inverses of relations

• Defining inverse functions

• Applying the horizontal line test

• Processes for finding inverses of functions

Inverses of Relations

• Definition: The inverse of a relation, denoted as $r^{-1}$, is found by switching the first and second coordinates of each ordered pair in the relation r.

  • If (a, b) is in r then (b, a) is in $r^{-1}$.

  • Example: If the relation r is given as { (4, -1), (-3, 2), (0, 5) }, then its inverse $r^{-1}$ is { (-1, 4), (2, -3), (5, 0) }.

• Explanation:

  • This switching of pairs mirrors concepts of symmetry but is distinct from it.

  • It is not correct to think of inverse relations as simply forming a function based on the original relation.

Inverse Functions

• An inverse function can only be derived from a function that passes the horizontal line test.

• Horizontal Line Test: A function passes this test if every horizontal line intersects its graph at most once.

  • If a function passes the horizontal line test, it can have an inverse that is also a function.

Example: Quadratic Function

• Original function: $y = x^2$

  • Inverse Relation: $x = y^2$ does not yield a valid inverse function because it fails the horizontal line test.

• Visual Representation:

  • The graph of $y = x^2$ does not pass the horizontal line test, showing duplicated y-values for different x-values.

One to One Functions

• Definition: A function is one to one (1-1) if for every distinct pair of elements $x<em>1$ and $x</em>2$ in the domain of f, the outputs are different: $f(x<em>1) eq f(x</em>2)$.

• Significance: Only one-to-one functions can have inverse functions.

• Vertical and Horizontal Line Tests Combined:

  • For a function to be one-to-one, it must pass both the vertical and horizontal line tests.

Inverse Relation vs. Inverse Function

• A function can have an inverse relation even when the inverse is not a function.

  • For example, the absolute value function $f(x) = |x|$ has an inverse relation but is not one-to-one as it fails the horizontal line test.

  • An example of a cubic function can be one-to-one, and thus its inverse can be a function.

Finding Inverses

Procedure for Finding Inverse Functions

1. Given a one-to-one function $f$ defined by $f(x)$,

2. Replace $f(x)$ with $y$ giving us the equation $y = f(x)$.

3. Solve this equation for $x$ in terms of $y$.

4. Switch variables to express the equation as $y = f^{-1}(x)$, thus defining the inverse function.

Example Processes

1. Example with a Cube Function:

   • Starting from the equation $y = (x - 1)^3 + 2$:

     • Replace: $x = (y - 1)^3 + 2$

     • Solve for $y$:

     • Subtract 2: $x - 2 = (y - 1)^3$

     • Take the cube root: $y - 1 = (x - 2)^{1/3}$

     • Rearranging gives: $y = (x - 2)^{1/3} + 1$, thus the inverse function is $f^{-1}(x) = (x - 2)^{1/3} + 1$.

2. Example with a Rational Function:

   • Start with $y= rac{x-3}{2x+1}$:

     • Switch: $x = rac{y - 3}{2y + 1}$

     • Solve for $y$, leading to the final form of inverse as shown in the earlier example.

Finding Inverses from Restricted Domains

• Functions that do not initially pass the one-to-one criteria can be altered by restricting their domain.

• Example: For $f(x) = x^2$, if we restrict the domain to $x ext{ from } [0, ext{positive infinity }]$, then we ensure it is one-to-one and has a valid inverse.

Composition of Functions and Their Inverses

• If $f$ and $f^{-1}$ are inverse functions, the following holds:

  1. $f(f^{-1}(x)) = x$ (for all $x$ in the domain of $f^{-1}$)

  2. $f^{-1}(f(x)) = x$ (for all $x$ in the domain of $f$)

• To demonstrate whether two functions are inverses, one can compute both compositions and see if they yield the identity function, producing $x$.

Summary and Conclusion

• The key to understanding inverse functions lies in recognizing the criteria for inverses to exist: the original function must be one-to-one, passing both vertical and horizontal line tests.

• If a function fails the horizontal line test, it may be adjusted through domain restrictions to allow for valid inverses.