Inverse Functions and Inverse Trigonometric Functions in Precalculus

Inverse Functions

Definition and Existence of Inverse Functions

An inverse function reverses the effect of the original function. For a function $f: A \to B$, the inverse $f^{-1}: B \to A$ exists only under certain conditions.

• Key Point 1: A function $f$ has an inverse if and only if it is one-to-one (injective).

• Key Point 2: If $f(x) = y$, then $f^{-1}(y) = x$.

• Key Point 3: If $f$ is not one-to-one, $f^{-1}$ cannot be defined for all $y$ in $B$.

Example: If $f(x) = x^2$ with domain $\mathbb{R}$, $f$ is not one-to-one because $f(4) = 16$ and $f(-4) = 16$. Thus, $f^{-1}$ is not defined for all $y$ in the range.

Graphical Representation

• Key Point: The graph of a function and its inverse are reflections across the line $y = x$.

• Example: For $f(x) = x^2$, the inverse $f^{-1}(y) = \sqrt{y}$ is only defined for $y \geq 0$ and $x \geq 0$ (the right branch of the parabola).

Inverse Trigonometric Functions

General Properties

Inverse trigonometric functions "undo" the effect of the original trigonometric functions, but their domains and ranges are restricted to ensure they are functions (one-to-one).

• Key Point 1: $\sin^{-1}(x)$ is not $\frac{1}{\sin x}$; it is the inverse function, not the reciprocal.

• Key Point 2: The inverse trigonometric functions are defined only on restricted intervals to ensure one-to-one correspondence.

Inverse Sine Function ($\sin^{-1} x$)

The inverse sine function, also called arcsin, is defined as follows:

• Definition: $y = \sin^{-1}(x) \iff x = \sin(y)$, where $-1 \leq x \leq 1$ and $-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$.

• Example: Find $\sin^{-1}\left(-\frac{1}{2}\right)$.

  • Let $\theta = \sin^{-1}\left(-\frac{1}{2}\right)$.

  • So $\sin(\theta) = -\frac{1}{2}$, with $-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}$.

  • Therefore, $\theta = -\frac{\pi}{6}$.

Inverse Cosine Function ($\cos^{-1} x$)

The inverse cosine function, or arccos, is defined as:

• Definition: $y = \cos^{-1}(x) \iff x = \cos(y)$, where $-1 \leq x \leq 1$ and $0 \leq y \leq \pi$.

• Example: Find $\cos^{-1}\left(-\frac{1}{2}\right)$.

  • Let $\theta = \cos^{-1}\left(-\frac{1}{2}\right)$.

  • So $\cos(\theta) = -\frac{1}{2}$, with $0 \leq \theta \leq \pi$.

  • Therefore, $\theta = \frac{2\pi}{3}$.

Inverse Tangent Function ($\tan^{-1} x$)

The inverse tangent function, or arctan, is defined as:

• Definition: $y = \tan^{-1}(x) \iff x = \tan(y)$, where $-\infty < x < \infty$ and $-\frac{\pi}{2} < y < \frac{\pi}{2}$.

• Example: Find $\tan^{-1}(-\sqrt{3})$.

  • Let $\theta = \tan^{-1}(-\sqrt{3})$.

  • So $\tan(\theta) = -\sqrt{3}$, with $-\frac{\pi}{2} < \theta < \frac{\pi}{2}$.

  • Therefore, $\theta = -\frac{\pi}{3}$.

Domain and Range Restrictions

• Key Point: The inverse trigonometric functions are only defined for certain values of $x$.

• Example: $\cos^{-1}(x)$ is only defined for $-1 \leq x \leq 1$. If $x$ is outside this interval, $\cos^{-1}(x)$ is not defined.

Compositions of Inverse Trigonometric Functions

Sometimes, compositions such as $\sin^{-1}(\sin(x))$ or $\cos^{-1}(\cos(x))$ are considered. The result depends on whether $x$ is in the principal range of the inverse function.

• Example: $\sin^{-1}(\sin(\frac{5\pi}{8}))$.

  • $\frac{5\pi}{8}$ is not in $[-\frac{\pi}{2}, \frac{\pi}{2}]$.

  • But $\sin(\frac{5\pi}{8}) = \sin(\frac{3\pi}{8})$.

  • Therefore, $\sin^{-1}(\sin(\frac{5\pi}{8})) = \frac{3\pi}{8}$.

Undefined Cases

• Key Point: If the input to an inverse trigonometric function is outside its domain, the function is not defined.

• Example: $\cos^{-1}(-2.4)$ is not defined because $-2.4$ is not in $[-1, 1]$.

• Example: $\cos^{-1}(\cos(\pi))$ is not defined because $\pi$ is not in the principal range $[0, \pi]$ for the composition.

Summary Table: Inverse Trigonometric Functions

Function	Definition	Domain	Range
$y = \sin^{-1}(x)$	$x = \sin(y)$	$-1 \leq x \leq 1$	$-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$
$y = \cos^{-1}(x)$	$x = \cos(y)$	$-1 \leq x \leq 1$	$0 \leq y \leq \pi$
$y = \tan^{-1}(x)$	$x = \tan(y)$	$-\infty < x < \infty$	$-\frac{\pi}{2} < y < \frac{\pi}{2}$

Additional info: The notes also include graphical illustrations of functions and their inverses, and examples of compositions and undefined cases, which are essential for understanding the behavior and limitations of inverse trigonometric functions in Precalculus.