Unit 4 of 8

Connecting Functions and Curves

  • 5 lessons
  • Parts 63 to 72

Calculus often builds new functions out of old ones and compares two curves on the same axes. This unit adds, multiplies, divides and composes functions, and finds inverse functions, restricting the domain when a function is not one-to-one.

It also finds where two curves meet and which one is higher between the meeting points, the question behind every area between curves. Circles and semicircles finish the unit, with the habit of choosing the right branch of a square root.

Every lesson will be a pair of videos, a theory part that explains the idea and a practice part that works through problems. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Combining and Inverting Functions

  1. Adding, Multiplying, and Dividing Functions

    How to add, subtract, multiply and divide functions by combining their outputs at the same input, and how to find the domain of the result, including the inputs a quotient must exclude.

    Theory part 63Practice part 64

  2. Composing Functions

    How to feed the output of one function into another, write the composite as one formula with correct parentheses, and find the inputs where the composite is defined.

    Theory part 65Practice part 66

  3. Inverse Functions and Domain Restrictions

    When a function can be reversed, how to build and verify its inverse, and how to restrict a function such as a shifted square so that a chosen branch has an inverse.

    Theory part 67Practice part 68

Comparing Curves

  1. Curve Intersections and Which Curve Is Higher

    How to find where two graphs meet by solving for equal outputs at a common input, how to reject false candidates, and how the sign of a difference shows which graph is higher.

    Theory part 69Practice part 70

  2. Circles, Semicircles, and Choosing a Branch

    Where the circle equation comes from, why a full circle is not the graph of a function, and how to split it into upper and lower semicircle functions with exact domains and ranges.

    Theory part 71Practice part 72