{{ drag.val }}

The "Undo" Machine

An inverse function does the exact opposite of the original function. Drag a number through to see!

{{ num }}
FUNCTION f(x)
Drop number here
{{ s1.midVal }}
INVERSE f⁻¹(x)
Drop output here
{{ s1.finalVal }}
The inverse perfectly undoes the function!

The Swap

To find an inverse algebraically, we first swap the inputs (x) and outputs (y).

Original Function:
{{ s2.leftVar }}
=
{{ s2.m }}
{{ s2.rightVar }}
{{ s2.b > 0 ? '+' : '-' }} {{ Math.abs(s2.b) }}
Variables swapped! Now solve for y:
x = {{ s2.m }}y {{ s2.b > 0 ? '+' : '-' }} {{ Math.abs(s2.b) }}
x {{ s2.b > 0 ? '-' : '+' }} {{ Math.abs(s2.b) }} = {{ s2.m }}y

Horizontal Line Test

A function only has an inverse if it passes the Horizontal Line Test. Scan the graph to see if the red line ever hits more than once.

Slide to scan
{{ s3Intersections.length > 1 ? 'Uh oh! Hits multiple points!' : 'Scanning...' }}

The Reflection

A function and its inverse are reflections over the line y = x. Plot the mirrored points!

y = x A ({{s4.p1.x}}, {{s4.p1.y}}) B ({{s4.p2.x}}, {{s4.p2.y}}) {{i===0 ? "A'" : "B'"}} ({{p.x}}, {{p.y}})
Tap the grid to plot A' (swap x and y of A) Great! Now plot B' Perfect reflection!
🎉

Master of Inverses!

You've learned how to undo functions, swap variables, check graphs, and reflect over y = x.