Mathematics — first biennium · Lesson
If f sends x to y, the inverse sends y back to x. To have an inverse on the whole domain, each output must come from only one input.
Prerequisites: Function concept and linear equations
Goal: Interpret the inverse as recovering the input and recognize the role of one-to-one behaviour in simple cases.
In this lesson: Interpret the inverse as recovering the input and recognize the role of one-to-one behaviour in simple cases.
New concept: The inverse swaps input and output: f(x)=y is equivalent to f⁻¹(y)=x. For the inverse to return a single value, each output of the original function must come from only one input on the chosen domain.
How to proceed: On the inverse graph the coordinates swap: (x,y) becomes (y,x).
You can already assume: Function concept and linear equations.
Move m to change the line's slope and direction; change q to move the y-intercept.
With a = 0 the curve loses this family's characteristic shape; also try a non-zero value.
The line graph makes the one-to-one behaviour of a linear function with non-zero slope visible.
Completed exercises: 0/5 · Correct on first attempt: 0
On the inverse graph the coordinates swap: (x,y) becomes (y,x).
Data / situation: f(x)=3x and f(1)=3.
ConceptStep separates what is required from what is recommended or useful as reinforcement. Only required prerequisites block your progress.
Relationships: solid line = required · dashed line = recommended · dotted line = reinforcement.