Functions, units and prerequisite repair

What this concept is

A function assigns one output to each permitted input. Its domain is part of its specification. Composition applies the inner function first. An inverse undoes a one-to-one function on a stated domain; it is not the reciprocal.

Why it matters here

Rates and densities have units. Algebra, logarithms and radians are dependencies, not material to guess while learning derivatives.

Concrete example

For f(x)=ln(x−2), the real domain is x>2. Its inverse is g(y)=2+exp(y), for all real y. Thus f(g(y))=y. A decay exp(−t/τ) requires t and τ in the same units so its exponent is dimensionless. If τ=4 seconds, half of the initial value remains at t=4 ln(2) seconds.

Common confusion / misconception

ln(a+b) is not ln(a)+ln(b). ln(ab)=ln(a)+ln(b) requires positive a,b. sin(π/2)=1 uses radians; degree inputs give different derivatives.

How to use it

First write the domain and units. Review x^a x^b=x^(a+b), exp(a+b)=exp(a)exp(b), ln(exp(x))=x, and exp(ln(x))=x for x>0. On the unit circle, angle θ in radians is arc length divided by radius; coordinates are (cos θ,sin θ). Memorize values at 0,π/2,π and the identity sin²θ+cos²θ=1. Sketch x²,1/x,exp(x),ln(x),sin(x) with domains before proceeding.

Check yourself

Why is the inverse of x² on all real numbers not a function? Restrict to x≥0 and give its inverse.

Mini drill or application

Solve exp(−3t)=1/4, restrict the domain of sqrt(5−x), and explain why ln(−1) is excluded here. Answers: ln(4)/3, x≤5, and no real number has exponential −1. If you cannot explain these, complete the prerequisite repair in the reading guide before lesson 2.

Read this only if stuck

OpenStax selective support. Use the reading guide for the section sequence and practice contract. Record your reasoning before consulting solutions.