Radicals as Inverse Polynomial Functions

Learning Outcomes

• Understand the definition of inverse functions.

• Recognize conditions under which radical and polynomial functions can be treated as inverses.

• Learn the process of finding the inverse of a polynomial function.

Inverse Functions

• Two functions, f and g, are considered inverse functions if for every coordinate pair (a, b) in f, there exists a corresponding pair (b, a) in g.

• The input and output of the inverse functions are interchanged.

• Only one-to-one functions have inverses that are also functions: A one-to-one function has a unique output value for every input value.

  • Passes the horizontal line test.

• If a function is not one-to-one, its domain can be restricted to make it one-to-one, thereby allowing the existence of an inverse function.

Tips for Success

• Work through examples by hand to internalize concepts.

• Repeat steps and explain them to yourself for clarity.

Example: Parabolic Trough

• A water runoff collector shaped like a parabolic trough can be analyzed using its depth.

• Coordinate system is defined, with the origin at the vertex of the parabola: The equation of the parabola is given by y(x) = ax².

• We must find the stretch factor a using the point (6, 18): From 18 = a(6²), we solve for a:

  18 = a(36)

  a = 18/36 = 1/2

• Thus, the equation becomes y(x) = (1/2)x².

• To find the width (w) of the water as a function of water depth (y): Solve for x: y = (1/2)x² leads to:

  2y = x²

  x = √(2y)

• Since this is not a one-to-one function, restrict x ≥ 0. Therefore:

  y = (x²)/2, x > 0

Surface Area Calculation

• The surface area (A) of the water is given by: For a trough length of 36 inches:

  A = l · w = 36(2x) = 72x

• This shows the relationship of the area based on the restricted function.

General Notes on Inverses

• Functions involving roots are often referred to as radical functions.

• Some polynomials have inverse functions, known as invertible functions.

• Denote inverse functions as f⁻¹(x): Important: f⁻¹(x) is different from the reciprocal of f(x) which is denoted as 1/f(x).

• Inverse functions “undo” each other: Formally:

  f⁻¹(f(x)) = x, f(f⁻¹(x)) = x

Verifying Inverse Functions

• To verify two functions are inverses:

  1. For functions f and g, show:

  g(f(x)) = x and f(g(x)) = x

How to Find Inverse Function of a Polynomial

1. Verify that the function is one-to-one.

2. Replace f(x) with y.

3. Interchange x and y.

4. Solve for y.

5. Rename found expression as f⁻¹(x).

Example: Finding the Inverse of a Simple Function

• Given f(x) = 1/(x + 1), find f⁻¹(x):

  1. Replace f(x) with y: y = 1/(x + 1).

  2. Interchange variables:

  x = 1/(y + 1)

  3. Solve for y:

  xy + x = 1 → xy = 1 - x → y = (1 - x)/x = f⁻¹(x)