Twelfth Grade · Math · week 1

Functions revisited: domain, range, and operations on functions

Determine domain and range with full algebraic rigor, and form the sum, difference, product, and quotient of two functions, including their domains.

Precalculus assumes the function fluency Algebra II built and immediately asks more of it. Every function still has a domain (the inputs it accepts) and a range (the outputs it produces), but this year you will find both for more demanding combinations -- and you will start combining whole functions the way you once combined numbers.

Given two functions f and g, you can build four new ones: (f+g)(x) = f(x)+g(x), (f-g)(x) = f(x)-g(x), (f*g)(x) = f(x)*g(x), and (f/g)(x) = f(x)/g(x). The domain of a sum, difference, or product is simply the overlap (intersection) of the two original domains -- both pieces have to be defined. A quotient adds one more requirement: you must also throw out any x where g(x) = 0, since that is still division by zero.

📋 For the teacher
Teach it (10–15 min):
  • Say plainly, every time: the domain of a combined function needs BOTH original domains to be satisfied, plus (for a quotient) the bottom function must not be zero.
  • Model evaluating f and g as two completely separate mini-steps before combining them -- rushing straight to one big computation is where sign and arithmetic errors creep in.
  • For even/odd, insist on writing out h(-x) fully as its own line before comparing it to h(x) or -h(x) -- guessing from the shape of the equation is not reliable yet.
  • Ask "could a function be both even and odd?" as a thought experiment: only f(x)=0 satisfies both f(-x)=f(x) and f(-x)=-f(x) at once, a sturdy fact worth having in your pocket.
Talk about it:
  • Why must the domain of f/g exclude every x where g(x)=0, even if that value causes no trouble in f?
  • What does it mean, graphically, for a function to be even? To be odd?
  • Could you build a function that is even everywhere except one point? What would that graph look like?

Mastery looks like: They correctly form and state the domain of a sum, difference, product, or quotient of two functions, and classify a function as even, odd, or neither using the algebraic test.

Common stumbles: Forgetting the extra exclusion (denominator = 0) when dividing two functions; declaring a function odd or even by its look instead of computing h(-x).

See it visually

Interactive, spoken-aloud tools that show this idea in motion.

MultiplicationEqual groups and arrays make products finally click.try →DivisionShare fairly and build equal groups.try →SubtractionTake away and count back, step by step.try →
Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
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