Step by step
The question
Question: graph f(x) = 3x + 3 and its inverse.
The inverse goes back: what comes out of f goes in, and what went in comes out.
Careful!
The little −1 is a name, not a power.
The inverse is not 1 over f.
1 Make a table for f
We pick −2, −1 and 0 for x and work out f.
For example, f of −1 is 0.
2 Swap x and y
For the inverse, every pair swaps: (−1, 0) becomes (0, −1).
What came out now goes in.
3 Draw f
Draw the grid.
Plot the points of f.
Join them: the line of f(x) = 3x + 3.
4 Draw the inverse
Now the swapped points.
Join them: the line of the inverse, f⁻¹.
5 The mirror line
Draw the line y = x.
The two lines are mirror images in it.
Fold the page along y = x, and f lands on f⁻¹.
6 Check
Let's check.
(0, 3) and (3, 0) are twins.
The line between them crosses y = x at its middle: the same distance on both sides.
The inverse lesson gives the rule f⁻¹(x) = (x − 3)/3.
Put in 3: we get 0.
So (3, 0) is on the red line.
It works!
Let's look back.
First: make a table for f.
Then: swap x and y.
Then: draw f.
Then: draw the inverse.
Then: the mirror line.
Last: check.