Worked examples
The question
Question: graph f(x) = 2x + 1 and its inverse.
f(x) = 2x + 1
The inverse goes back: what comes out of f goes in, and what went in comes out.
inverse: swaps in and out
Careful!
The little −1 is a name, not a power.
The inverse is not 1 over f.
Careful: f −1(x) is not 1f(x)
1 Make a table for f
We pick −1, 0 and 1 for x and work out f.
For example, f of 0 is 1.
2 Swap x and y
For the inverse, every pair swaps: (0, 1) becomes (1, 0).
What came out now goes in.
3 Draw f
Draw the grid.
Plot the points of f.
Join them: the line of f(x) = 2x + 1.
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.
(1, 3) and (3, 1) 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 − 1)/2.
Put in 3: we get 1.
So (3, 1) is on the red line.
It works!
f −1(3) = 3 − 12 = 22 = 1
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.
▶ Watch it solved
The question
Question: graph f(x) = 3x − 2 and its inverse.
f(x) = 3x − 2
The inverse goes back: what comes out of f goes in, and what went in comes out.
inverse: swaps in and out
Careful!
The little −1 is a name, not a power.
The inverse is not 1 over f.
Careful: f −1(x) is not 1f(x)
1 Make a table for f
We pick −1, 0 and 1 for x and work out f.
For example, f of 0 is −2.
2 Swap x and y
For the inverse, every pair swaps: (0, −2) becomes (−2, 0).
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 − 2.
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.
(−1, −5) and (−5, −1) 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 + 2)/3.
Put in −5: we get −1.
So (−5, −1) is on the red line.
It works!
f −1(−5) = −5 + 23 = −33 = −1
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.
▶ Watch it solved
The question
Question: graph f(x) = x/2 + 1 and its inverse.
f(x) = x2 + 1
The inverse goes back: what comes out of f goes in, and what went in comes out.
inverse: swaps in and out
Careful!
The little −1 is a name, not a power.
The inverse is not 1 over f.
Careful: f −1(x) is not 1f(x)
1 Make a table for f
We pick −2, 0 and 2 for x and work out f.
For example, f of 0 is 1.
2 Swap x and y
For the inverse, every pair swaps: (0, 1) becomes (1, 0).
What came out now goes in.
3 Draw f
Draw the grid.
Plot the points of f.
Join them: the line of f(x) = x/2 + 1.
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.
(−2, 0) and (0, −2) 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) = 2(x − 1).
Put in 0: we get −2.
So (0, −2) is on the red line.
It works!
f −1(0) = 2(0 − 1) = 2(−1) = −2
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.
▶ Watch it solved
