Inverse Trig Functions Notes

Inverse Functions Review

Definition

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

One-to-One Functions

A function ff is one-to-one if f(x<em>1)≠f(x</em>2)f(x<em>1) ≠ f(x</em>2) whenever x<em>1≠x</em>2x<em>1 ≠ x</em>2. 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 ff is one-to-one with domain AA and range BB, then f−1(y)=x⇔f(x)=yf^{-1}(y) = x ⇔ f(x) = y for any yy in BB.

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

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

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

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

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

Derivatives of Inverse Functions

Theorem

If ff is a one-to-one continuous function on an interval II, then:

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

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

Theorem

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

Inverse Trig Functions and Their Derivatives

Derivatives of Inverse Trigonometric Functions

Let uu be a differentiable function of xx.

  • ddx[arcsin⁡u]=u′1−u2\frac{d}{dx}[\arcsin u] = \frac{u'}{\sqrt{1 - u^2}}

  • ddx[arccos⁡u]=−u′1−u2\frac{d}{dx}[\arccos u] = -\frac{u'}{\sqrt{1 - u^2}}

  • ddx[arctan⁡u]=u′1+u2\frac{d}{dx}[\arctan u] = \frac{u'}{1 + u^2}

  • ddx[arccot⁡u]=−u′1+u2\frac{d}{dx}[\operatorname{arccot} u] = -\frac{u'}{1 + u^2}

  • ddx[arcsec⁡u]=u′∣u∣u2−1\frac{d}{dx}[\operatorname{arcsec} u] = \frac{u'}{\left| u \right| \sqrt{u^2 - 1}}

  • ddx[arccsc⁡u]=−u′∣u∣u2−1\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⁡xf(x) = \sin x restricted domain:

  • f(x)=cos⁡xf(x) = \cos x restricted domain:

  • f(x)=tan⁡xf(x) = \tan x restricted domain:

Key Limits

lim⁡<em>x→∞arctan⁡x=\lim<em>{x \to \infty} \arctan x = lim⁡</em>x→−∞arctan⁡x=\lim</em>{x \to -\infty} \arctan x =