## Functions

Hereโs an โingeliousโ way on H2 Mathematics Functions summary.

**One-to-One **

$f(x)$ = paint white balls to blue

When x = white, $f(x)$ = blue

**One-to-One (inverse function)**

$f^{-1}(x)$ = paint blue balls to white

When x = blue, $f^{-1}(x)$ = white

**Many-to-One **

$f(x)$ = paint red or blue balls to white

When x = red or blue, $f(x)$ = white

**One-to-Many (no inverse function)**

When x = white, $f^{-1}(x)$ = ??

**Composite Functions **

$f(x)$ = paint white balls to blue

When x = white, $f(x)$ = blue

$g(x)$ = paint blue balls to red

When x = shades of blue, $g(x)$ = red

$gf(x)$ = paint white balls to red

When x = white, $gf(x)$ = red

$gf(x) \neq fg(x) $

$D_f$ = [white balls]

$D_g$ = [shades of blue balls]

$D_{gf}$= [white balls]

$R_{gf}$ = [red balls]

**Conditions for Composite Functions **

If $f(x)$ changes, such that

$f(x)$ = paint white balls to orange

When x = white, $f(x)$ = orange

$g(x)$ = paint blue balls to red

$gf(x)$ = paint white balls to red

When x = white, $gf(x)$ = undefined

**Domain and Range for Composite Functions **

$f(x)$ = paint white balls to orange

When x = white, $f(x)$ = shades of blue

$g(x)$ = paint blue balls to shades of red

When x = shades of blue, $g(x)$ = shades of red

$D_g$ can take in 4 shades of blue balls.

$R_g$ produces 4 shades of corresponding shades of red balls.

Max $R_g$ is 4 shades of red balls.

If $R_f$ < $D_g$ , range of $R_{gf}$ will be smaller than range of $R_g$.

In this case, $R_{gf}$ includes only 2 shades of red.

$R_{gf}$ can be smaller than range of $R_g$ , if $R_f < D_g$