Inverse Functions: Index

I’m doing something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on the different definitions on inverse functions that appear in Precalculus and Calculus.

Square Roots, nth Roots, and Rational Exponents

Part 1: Simplifying $\sqrt{x^2}$

Part 2: The difference between $\sqrt{t}$ and solving $x^2 = t$

Part 3: Definition of an inverse function and the horizontal line test

Part 4: Why extraneous solutions may occur when solving algebra problems involving a square root

Part 5: Defining $\sqrt{x}$

Part 6: Consequences of the definition of $\sqrt{x}$: simplifying $\sqrt{x^2}$

Part 7: Defining $\sqrt[n]{x}$ if $n$ is odd or even

Part 8: Rational exponents if the denominator of the exponent is odd or even

Arcsine

Part 9: There are infinitely many solutions to $\sin x = 0.8$

Part 10: Defining arcsine with domain $[-\pi/2,\pi/2]$

Part 11: Pedagogical thoughts on teaching arcsine.

Part 12: Solving SSA triangles: impossible case

Part 13: Solving SSA triangles: one way of getting a unique solution

Part 14: Solving SSA triangles: another way of getting a unique solution

Part 15: Solving SSA triangles: continuation of Part 14

Part 16: Solving SSA triangles: ambiguous case of two solutions

Part 17: Summary of rules for solving SSA triangles

Arccosine

Part 18: Definition for arccosine with domain $[0,\pi]$

Part 19: The Law of Cosines and solving SSS triangles

Part 20: Identifying impossible triangles with the Law of Cosines

Part 21: The Law of Cosines provides an unambiguous angle, unlike the Law of Sines

Part 22: Finding the angle between two vectors

Part 23: A proof for why the formula in Part 22 works

Arctangent

Part 18: Definition for arctangent with domain $(-\pi/2,\pi/2)$

Part 24: Finding the angle between two lines

Part 25: A proof for why the formula in Part 24 works.

Arcsecant

Part 26: Defining arcsecant using $[0,\pi/2) \cup (\pi/2,\pi]$

Part 27: Issues that arise in calculus using the domain $[0,\pi/2) \cup (\pi/2,\pi]$

Part 28: More issues that arise in calculus using the domain $[0,\pi/2) \cup (\pi/2,\pi]$

Part 29: Defining arcsecant using $[0,\pi/2) \cup [pi,3\pi/2)$

Logarithm

Part 30: Logarithms and complex numbers