Inverse Functions and Their Properties

Definition and Graphical Interpretation

Inverse functions are fundamental in mathematics, allowing us to reverse the effect of a function. For a function f to have an inverse, it must be one-to-one, meaning each output corresponds to exactly one input. The graph of an inverse function f-1 is the reflection of the graph of f across the line y = x.

• One-to-one function: Each x-value maps to a unique y-value, and vice versa.

• Inverse function: If f is one-to-one, then f-1 exists such that f(f-1(x)) = x and f-1(f(x)) = x.

• Domain and range: The domain of f becomes the range of f-1 and vice versa.

• Graphical reflection: The point (a, b) on f corresponds to (b, a) on f-1.

Steps to find an inverse function:

1. Replace f(x) with y and interchange x and y.

2. Solve for y.

3. Replace y with f-1(x).

Restricting Domains to Obtain Inverses

Horizontal Line Test and Domain Restriction

Not all functions are one-to-one over their entire domains. The horizontal line test determines if a function has an inverse: if any horizontal line crosses the graph more than once, the function is not one-to-one. By restricting the domain, we can often obtain a one-to-one function and thus define an inverse.

• Example: The sine function y = \sin x is not one-to-one on (-\infty, \infty), but restricting the domain to \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] makes it one-to-one.

Inverse Sine Function (arcsin)

Definition, Domain, and Range

The inverse sine function is denoted as y = \sin^{-1} x or y = \arcsin x. It returns the angle y in the interval \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] whose sine is x.

• Domain: [-1, 1]

• Range: \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]

• Odd function: \sin^{-1}(-x) = -\sin^{-1}(x)

Examples:

• \( y = \arcsin \frac{1}{2} = \frac{\pi}{6} \)

• \( y = \sin^{-1}(-1) = -\frac{\pi}{2} \)

• \( y = \sin^{-1}(-2) \) is undefined (since -2 is outside the domain).

Inverse Cosine Function (arccos)

Definition, Domain, and Range

The inverse cosine function is denoted as y = \cos^{-1} x or y = \arccos x. It returns the angle y in the interval [0, \pi] whose cosine is x.

• Domain: [-1, 1]

• Range: [0, \pi]

• Not symmetric with respect to the origin or y-axis.

Examples:

• \( y = \arccos 1 = 0 \)

• \( y = \cos^{-1} \left(-\frac{\sqrt{2}}{2}\right) = \frac{3\pi}{4} \)

Inverse Tangent Function (arctan)

Definition, Domain, and Range

The inverse tangent function is denoted as y = \tan^{-1} x or y = \arctan x. It returns the angle y in the interval \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) whose tangent is x.

• Domain: (-\infty, \infty)

• Range: \left( -\frac{\pi}{2}, \frac{\pi}{2} \right)

• Odd function: \tan^{-1}(-x) = -\tan^{-1}(x)

• Horizontal asymptotes: y = \frac{\pi}{2}, y = -\frac{\pi}{2}

Examples:

• \( y = \arctan 1 = \frac{\pi}{4} \)

• \( y = \tan^{-1}(-1) = -\frac{\pi}{4} \)

Inverse Cotangent, Secant, and Cosecant Functions

Definitions, Domains, and Ranges

• Inverse Cotangent: y = \cot^{-1} x or y = \text{arccot} x

  • Domain: (-\infty, \infty)

  • Range: (0, \pi)

  • Horizontal asymptotes: y = 0, y = \pi

• Inverse Cosecant: y = \csc^{-1} x or y = \text{arccsc} x

  • Domain: (-\infty, -1] \cup [1, \infty)

  • Range: \left[ -\frac{\pi}{2}, 0 \right) \cup \left( 0, \frac{\pi}{2} \right]

  • Odd function: \csc^{-1}(-x) = -\csc^{-1}(x)

• Inverse Secant: y = \sec^{-1} x or y = \text{arcsec} x

  • Domain: (-\infty, -1] \cup [1, \infty)

  • Range: [0, \frac{\pi}{2}) \cup (\frac{\pi}{2}, \pi]

  • Horizontal asymptote: y = \frac{\pi}{2}

Summary Table: Inverse Trigonometric Functions

Evaluating Inverse Trigonometric Functions

Calculator Use and Exact Values

• Calculator keys typically provide sin-1, cos-1, and tan-1.

• Other inverses can be computed as:

  • cot-1 x = π/2 – tan-1 x

  • sec-1 x = cos-1 (1/x)

  • csc-1 x = sin-1 (1/x)

• Always ensure x is in the correct domain for the function.

Inverse Trigonometric Values as Angles

Inverse trigonometric functions return angles in radians, but sometimes degree measures are required. For example:

• \( \theta = \arctan 1 = \frac{\pi}{4} = 45^\circ \)

• \( \theta = \sin^{-1}(-\frac{1}{2}) = -\frac{\pi}{6} = -30^\circ \)

• \( \theta = \cos^{-1} \frac{1}{2} = \frac{\pi}{3} = 60^\circ \)

Compositions and Applications

Finding Exact Values Using Definitions

Inverse trigonometric functions can be used in composition with other trigonometric functions to find exact values. For example:

• \( \sin(\tan^{-1}(\frac{3}{2})) \)

• \( \tan(\cos^{-1}(-\frac{5}{13})) \)

• \( \cos(\arctan 3 + \arcsin \frac{1}{3}) \)

• \( \tan(2 \arcsin \frac{2}{5}) \)

These can be solved by drawing right triangles and using the definitions of the trigonometric functions.