Inverse Trig Functions Notes

Inverse Functions Review

Definition

A function g is the inverse of f if f(g(x)) = x for all x in the domain of g AND g(f(x)) = x for all x in the domain of f. The inverse of f is denoted as f^{-1}.

One-to-One Functions

A function f is one-to-one if f(x1) ≠ f(x2) whenever x1 ≠ x2. A one-to-one function passes the horizontal line test (no horizontal line intersects the graph more than once).

Key Properties of Inverse Functions

  1. If f is one-to-one with domain A and range B, then f^{-1}(y) = x ⇔ f(x) = y for any y in B.

  2. Domain of f^{-1} = Range of f, Range of f^{-1} = Domain of f

  3. If a function has an inverse, then the inverse is unique.

  4. (a, b) is on the graph of f if and only if (b, a) is on the graph of f^{-1}.

  5. The graph of f^{-1} is a reflection of the graph of f across the line y = x.

  6. Cancellation Equations: f(f^{-1}(x)) = x and f^{-1}(f(x)) = x

Derivatives of Inverse Functions

Theorem

If f is a one-to-one continuous function on an interval I, then:

  1. f^{-1} is continuous on its domain.

  2. If f is differentiable on an interval containing c and f'(c) ≠ 0, then f^{-1} is differentiable at f(c).

Theorem

Let f be differentiable on an interval I. If f has an inverse function g, then g is differentiable at any x for which f'(g(x)) ≠ 0. Moreover,
g'(x) = \frac{1}{f'(g(x))}, where f'(g(x)) ≠ 0

Inverse Trig Functions and Their Derivatives

Derivatives of Inverse Trigonometric Functions

Let u be a differentiable function of x.

  • \frac{d}{dx}[\arcsin u] = \frac{u'}{\sqrt{1 - u^2}}

  • \frac{d}{dx}[\arccos u] = -\frac{u'}{\sqrt{1 - u^2}}

  • \frac{d}{dx}[\arctan u] = \frac{u'}{1 + u^2}

  • \frac{d}{dx}[\operatorname{arccot} u] = -\frac{u'}{1 + u^2}

  • \frac{d}{dx}[\operatorname{arcsec} u] = \frac{u'}{\left| u \right| \sqrt{u^2 - 1}}

  • \frac{d}{dx}[\operatorname{arccsc} u] = -\frac{u'}{\left| u \right| \sqrt{u^2 - 1}}

Domain Restrictions for Inverse Trig Functions

Trig functions are not one-to-one, so domain restrictions are necessary to define inverse trig functions.

  • f(x) = \sin x restricted domain:

  • f(x) = \cos x restricted domain:

  • f(x) = \tan x restricted domain:

Key Limits

\lim{x \to \infty} \arctan x = \lim{x \to -\infty} \arctan x =