Transcendental Functions

The condition for an inverse to exist is that you don't "lose information" when applying the function: no two distinct points are mapped to the same point.

Definition 1: One-to-one
A function \(f\) is said to be one-to-one if it maps different points to different points. Symbolically: \(x_1 \neq x_2 \implies f(x_1) \neq f(x_2) \).

Definition 2: Inverse function
If \(f\) is one-to-one then it has an inverse, written \(f^{-1}\). The value of \(f^{-1}(y)\) is defined to be the number \(x\) in the domain of \(f\) for which \(f(x) = y\).

Differentiating inverse functions
As a consequence of the second definition we get the identities \(f^{-1}(f(x)) = x\) and \(f(f^{-1}(x)) = x\) when an inverse exists. Suppose the domain of \(f\) is an interval, that \(f'\) exists and doesn't change sign on the interval. Then \(f\) is either increasing or decreasing on that interval, therefore one-to-one, and thus there exists an inverse \(f^{-1}\). Assuming that \(f^{-1}\) is differentiable, we can find its derivative by differentiating both sides of the above identity:

\[f'(f^{-1}(x)) \cdot \frac{d}{dx} f^{-1}(x) = 1, \] implying \[ \frac{d}{dx} f^{-1}(x) = \frac{1}{f'(f^{-1}(x))}. \]

(Actually one doesn't have to assume that \(f^{-1}\) is differentiable, as one can show that the stated assumption on \(f\) will automatically imply it.)

The following are some of the most important properties of exponential and logarithmic functions, and should be learned by heart. The results are stated for a general base \(a>0\), but in practice one mostly uses the bases 2, \(e\) and 10.

Laws of exponents
Let \(a>0\). Then \[ \begin{align} a^0 &= 1, & \qquad a^{x+y} &= a^x a^y, \\[1em] a^{-x} &= \frac{1}{a^x}, & \qquad a^{x-y} &= \frac{a^x}{a^y}, \\[1em] (a^x)^y &= a^{xy}, & \qquad (ab)^x &= a^x b^x. \end{align} \]

Laws of logarithms
Let \(x, y, a, b>0\) and \(a,b \neq 1\). Then \[ \begin{align} \log_a 1 &= 0, & \qquad \log_a(xy) &= \log_a x + \log_a y, \\[1em] \log_a\left(\frac{1}{x}\right)&=-\log_a x, & \qquad \log_a \left(\frac{x}{y}\right) &= \log_a x-\log_a y. \\[1em] \log_a (x^y) &= y \log_a x, & \qquad \log_a x &= \frac{\log_b x}{\log_b a}. \end{align} \]

Important limits
Let \(a > 1\). Then \[ \begin{align} &\lim_{x\to -\infty}a^x = 0, & \qquad \lim_{x\to \infty}a^x = \infty, \\[1em] &\lim_{x\to 0+} \log_a x = -\infty, &\qquad \lim_{x\to \infty} \log_a x = \infty. \end{align} \]

Derivatives
Both the exponential and logarithm functions are differentiable, with derivatives \[\frac{d}{dx} a^x = \ln(a) a^x, \] \[\frac{d}{dx} \log_a x = \frac{1}{x \ln(a)}.\]

Since the trigonometric functions are periodic, they are not one-to-one on the entire real line. One therefore has to restrict their domains suitably if one wants an inverse function.

Definition 9: The inverse sine function
The inverse of \(\sin\), called \(\arcsin\), is defined by \[y = \arcsin(x) \Longleftrightarrow x = \sin(y) \ \mathrm{and} \ -\pi/2 \le y \le \pi/2.\] (\(\arcsin\) is the inverse of the sine function with its domain restricted to \([-\pi/2,\pi/2]\).)

Definition 11: The inverse tangent function
The inverse of \(\tan\), called \(\arctan\), is defined by \[y = \arctan(x) \Longleftrightarrow x = \tan(y) \ \mathrm{and} \ -\pi/2 < y < \pi/2.\] (\(\arctan\) is the inverse of the tangent function with its domain restricted to \((-\pi/2,\pi/2)\).)

Definition 12: The inverse cosine function
The inverse of \(\cos\), called \(\arccos\), is defined by \[y = \arccos(x) \Longleftrightarrow x = \cos(y) \ \mathrm{and} \ 0 \le y \le \pi.\] (\(\arccos\) is the inverse of the cosine function with its domain restricted to \([0,\pi]\)). An equivalent definition is \(\arccos(x) = \pi/2-\arcsin(x)\).

Derivatives
\[ \frac{d}{dx} \arcsin(x) = \frac{1}{\sqrt{1-x^2}} \] \[ \frac{d}{dx} \arccos(x) = \frac{-1}{\sqrt{1-x^2}} \] \[ \frac{d}{dx} \arctan(x) = \frac{1}{1+x^2} \]

The hyperbolic functions are combinations of exponentials, and as such natural counterparts of their trigonometric kin; each family of functions can be obtained from the other by a complex shift of variables, and—up to sign changes—all formulas for the trigonometric functions are valid also for their hyperbolic counterparts.

Definition 15: The hyberbolic cosine and hyperbolic sine functions
\[ \cosh(x) = \frac{e^x+e^{-x}}{2} \]

\[ \sinh(x) = \frac{e^x-e^{-x}}{2} \]

Definition 17: The hyperbolic tangent function
\[ \tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{e^x-e^{-x}}{e^x+e^{-x}} \]

Derivatives
\[ \frac{d}{dx} \sinh(x) = \cosh(x) \] \[ \frac{d}{dx} \cosh(x) = \sinh(x) \] \[ \frac{d}{dx} \tanh(x) = \frac{1}{\cosh^2(x)} \]

N.b. It can be shown that \(\sin(x) = \frac{1}{2i} (e^{ix} - e^{-ix})\) and \(\cos(x) = \frac{1}{2} (e^{ix} + e^{-ix})\), explaining the relationship between the trigonometric and hyperbolic functions.