Composing Rules
Part of the Functions & Transformations section of Coddy's Algebra 2 journey. Lesson 24 of 58.
A composition can be written as one new rule. Take p(x) = x + 1 and q(x) = x^2 + 4. In q(p(x)), the output of p is the expression x + 1, and it goes into q in place of x, whole and in brackets: q(p(x)) = (x + 1)^2 + 4.
Then multiply out and collect: (x + 1)^2 is x^2 + 2x + 1, so the rule is x^2 + 2x + 5.
Check with one number. At x = 2, p(2) is 3 and q(3) is 13. The new rule gives 4 + 4 + 5, which is 13 too. One rule now does the work of two.
Challenge
For f(x) = x + 3 and g(x) = x^2 - 2, find g(f(x)) as one rule, multiplied out and collected.
Try it yourself
Multiply everything out
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Functions & Transformations
4Combining Functions
Adding FunctionsSubtracting FunctionsMultiplying FunctionsAt a Single NumberRecap: Combining10Functions in Stories
Writing the RuleUsing the RuleWorking BackwardUndoing a ConversionRecap: Stories2Expressions as Inputs
A Letter as InputA Sum as InputShifting the InputSolving f(x) = kRecap: Inputs as Expressions5Composition
One Inside AnotherOrder MattersA Function of ItselfComposing RulesBreaking ApartRecap: Composition8Stretching and Flipping
Taller and FlatterFlipping OverFlip, Then ShiftReading the RuleRecap: Stretch and Flip