Notes on Invertibility, Monotonicity, and Chain Rule

Invertibility and the Horizontal Line Test

  • Invertibility (on a domain) means the function is one-to-one (injective): each output y in the image f(D) comes from a unique input x in D.
  • Horizontal line test: A graph passes the test on domain D if no horizontal line y = c intersects the graph at more than one point with x in D. If some y has two or more preimages in D, the function is not invertible on D.
  • Transcript intuition:
    • If you can piece together a graph or know the function is, or behaves like, an odd-power shape, you might expect it to be invertible on a broad domain (heuristic): for example, f(t) = t^3 is strictly increasing and invertible on all of R.
    • Even without a formula or full graph, you can reason about invertibility by thinking about monotonic segments or how a function could fail the horizontal line test.
  • Important subtlety from the transcript:
    • A horizontal tangent line does not by itself signal non-invertibility. If the horizontal line is tangent and intersects the graph at only one point within the considered domain, the function can still be invertible on that domain.
    • What matters for non-invertibility is whether there exists some y such that the equation f(x) = y has more than one solution x in the domain. In other words, you need a sign change in the derivative (or a true non-monotone behavior) over the domain to cause multiple preimages for some y.
  • Example intuition: consider a graph restricted to a segment where the curve is strictly increasing or strictly decreasing; on that restricted domain it is invertible even if the full graph is not.

Monotonicity, Tangents, and Domain Restrictions

  • A function that goes up and then down typically has a derivative that changes sign: it increases (f' > 0) on a stretch and then decreases (f' < 0) after a turning point.
  • Concavity intuition:
    • The transcript states the second derivative is always negative (h''(x) < 0), which means the graph is concave down.
    • With a finite positive velocity initially (f'(x) > 0 at first), the derivative eventually becomes negative, so the function increases then decreases.
    • This is summarized as: the function goes up, and then it goes down.
  • Domain restriction for invertibility:
    • A non-monotone function can still be invertible if we restrict its domain to an interval where it is strictly monotone (e.g., where f' does not change sign).
  • Tangent observations and invertibility:
    • A horizontal tangent at a local maximum/minimum may exist, but the function can still be invertible on a restricted domain if the horizontal tangent does not create multiple intersections with the same y-value within that domain.

The Chain Rule for Composition

  • Core rule:
    • If y = g(h(x)), then the derivative is:
    • (g∘h)′(x)=g′(h(x)) h′(x).(g \circ h)'(x) = g'(h(x))\, h'(x).
  • Intuition:
    • The outer function's rate is evaluated at the inner function’s value, multiplied by the rate of change of the inner function.

Connections and Implications

  • Inverse functions:
    • Invertibility on a domain is a prerequisite for the existence of an inverse function on that domain.
    • If f is not globally invertible, you can often obtain an inverse by restricting the domain to a monotone interval where f is injective.
  • Derivative sign and monotonicity:
    • If f'(x) > 0 for all x in D, f is strictly increasing on D.
    • If f'(x) < 0 for all x in D, f is strictly decreasing on D.
    • If f' changes sign on D, the function is not globally monotone on D, but may be monotone on subintervals.
  • Practical relevance:
    • In modeling real-world processes (e.g., height over time), monotone segments allow a unique mapping from height to time on those segments, which underpins the idea of an inverse relationship on that segment.

Examples and Scenarios

  • Odd-power heuristic (illustrative):
    • Example: f(t) = t^3 is strictly increasing on all of (\mathbb{R}), hence invertible with inverse f^{-1}(y) = \sqrt[3]{y}.
  • Concave-down motion (non-global invertibility):
    • Example: ( h(x) = -x^2 + 4x ) has ( h'(x) = -2x + 4 ) and ( h''(x) = -2 < 0 ).
    • It increases on ( (-\infty, 2] ) and decreases on ([2, \infty)). On the full domain, it is not invertible, but it is invertible on each monotone piece, e.g., ( (-\infty, 2] ) or ([2, \infty) ).

Summary of Key Formulas and Concepts

  • Invertibility criterion (horizontal line test):
    • A function f is invertible on a domain D if and only if for every y in f(D), the equation f(x) = y has at most one solution x in D.
  • Monotonicity and derivatives:
    • If (f'(x) > 0) for all x in D, then f is strictly increasing on D.
    • If (f'(x) < 0) for all x in D, then f is strictly decreasing on D.
    • If (f') changes sign on D, f is not globally monotone on D (but may be monotone on subintervals).
  • Chain rule for composition:
    • ddx(g(h(x)))=g′(h(x))  h′(x).\frac{d}{dx} \bigl(g(h(x))\bigr) = g'(h(x)) \; h'(x).
  • Practical takeaway:
    • To ensure a unique mapping from outputs back to inputs, restrict the domain to a monotone interval if the full function is not monotone.