Swap and Solve
Part of the Functions & Transformations section of Coddy's Algebra 2 journey. Lesson 28 of 58.
To find an inverse rule, trade the roles of input and output. Take f(x) = 4x + 8 and write it with y for the output: y = 4x + 8. Then swap x and y: x = 4y + 8. This equation says the same thing with input and output traded.
Now solve for y, since y is the new output. Subtract 8: x - 8 = 4y. Divide by 4 and split the fraction: y = x/4 - 2. So f^-1(x) = x/4 - 2.
Check it with one number. f(3) is 20, and the inverse sends 20 to 5 minus 2, which is 3. The swap is the whole idea of an inverse; the solving after it is ordinary algebra.
Challenge
Find the inverse of f(x) = 2x + 6. The board already holds the swapped equation x = 2y + 6: solve it for y, and write the result multiplied out.
Try it yourself
Solve for y
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 Flip6Inverse Functions
Undoing a FunctionSwap and SolveFractions in InversesChecking an InverseA Value of the InverseRecap: Inverses