Self-inverse functions alternate by parity

f

Total marks: 10

Solution

(i)

Self-inverse functions

$f(x)=\dfrac{ax+b}{cx-2}$ for $x\ne\frac2c$ . By finding $f^{-1}(x)$ , show that $a=2$ for $f$ to be self-inverse.

Explore this idea

Hint

Write $y=\frac{ax+b}{cx-2}$ and make $x$ the subject.
Compare $f^{-1}$ with $f$ .

Answer · (i)

f^{-1}(x)=(2x+b)/(cx-a)

Explore the idea

A self-inverse graph is its own mirror image

With a = 2, b = 3, c = 5 the curve and its inverse lie on top of each other. One branch is shown (x > 2/5).

Point on f: input xx = 1.00

Reflection swaps the coordinates. The segment PQ is perpendicular to y = x, and its blue midpoint M lies on that line.

P = (x, f(x))

(1, 1.66667)

Q = (f(x), x)

(1.66667, 1)

on the inverse

Domain of f

x>\frac25

(right branch)

Range of f = domain of f⁻¹

y>\frac25

Midpoint M on y = x

(1.33333, 1.33333)

F
F⁻¹, which is the same curve
Y = x
PQ is perpendicular to y = x; its blue midpoint M lies on that line

Key idea

A self-inverse graph is symmetric about $y=x$ : the reflection of any point on it is also on it.

Ideas to try

Move

x

and compare the coordinates of

P

and

Q

: are they on the same curve?

(ii)

Powers of a function

Now $a=2$ , $b=3$ , $c=5$ . Find $f^2(x)$ .

Hint

$f=f^{-1}$ , so $f^2=ff^{-1}$ .

Answer · (ii)

f^2(x)=x

(iii)

Reducing large powers

Evaluate $f^{71}(4)$ .

Explore this idea

Hint

$f^2$ is the identity, so only the parity of the power matters.
$71=2\times35+1$ .

Answer · (iii)

f^{71}(4)=11/18

Explore the idea

Apply f to 4, again and again

The value alternates between 4 and 11/18, because f applied twice returns the input.

Start: Start at 4

Start with the input $4$ .

Marked points at this step, rounded to 6 significant figures.
Point	x	y
4	4	4

Number line shown as y = x
Current value

Ideas to try

What is

f^{70}(4)

(iv)

Existence of composites

$g(x)=2x^2-3$ for all real $x$ . Explain why $fg$ does not exist.

Explore this idea

Hint

$fg$ exists when $R_g\subseteq D_f$ .
Find the one value excluded from $D_f$ and check whether $g$ reaches it.

Answer · (iv)

fg

Explore the idea

Does g ever output 2/5?

The input y = k to f is bad when k = 2/5. The parabola reaches that height at two places.

Height of the horizontal linek = 0.4

Try an example

Test boundary heights

Buttons use full-precision boundary values; numbers shown here are rounded.

Number of distinct intersections

Two intersections. In particular, $g$ reaches $2/5$ , an excluded input of $f$ .

All listed intersections are within the original drawing window. A touching point counts once; hollow endpoints are excluded.

Coordinates are rounded to 6 significant figures. Inspect a row to highlight that intersection and bring it into view.
Point	x	y
	-1.30384	0.4
	1.30384	0.4

G
Y = k
Inputs where g takes the value k
Intersection selected in the coordinate table

Key idea

Because $g$ can output $\frac25$ , feeding it to $f$ would divide by zero.

Ideas to try

Set

k=\frac25

and read off the two inputs that make

g=\frac25

(v)

Solving rational equations

Find the exact solutions of $gf(x)=5$ .

Hint

$gf(x)=2f(x)^2-3$ .
Solve $f(x)^2=4$ , so $f(x)=\pm2$ .

Answer · (v)

x=7/8

Self-inverse functions alternate by parity

Solution

Self-inverse functions

Clear the fraction

Collect $x$

Inverse

Compare with $f$

A self-inverse graph is its own mirror image

Powers of a function

Use $f=f^{-1}$

Reducing large powers

Reduce

Evaluate

Apply f to 4, again and again

Existence of composites

Range of $g$

Domain of $f$

A bad output

Does g ever output 2/5?

Solving rational equations

Set up

Take both square roots

Solve each

Clear the fraction

Collect xxx

Inverse

Compare with fff

Use f=f−1f=f^{-1}f=f−1

Reduce

Evaluate

Range of ggg

Domain of fff

A bad output

Set up

Take both square roots

Solve each

Collect $x$

Compare with $f$

Use $f=f^{-1}$

Range of $g$

Domain of $f$