Question

Write two functions 𝑢(𝑥) and 𝑣(𝑥)?

How would you split up this function to have two different equations?

Answer 1

See below.

Explanation:

There are infinite answers to this question. Putting

`{(u(x) = (a x+b)/(cx + d)),(v(x) = e x^2+f x + g):}` we have

`u(v(x))=(b + a (g + f x + e x^2))/(d + c (g + f x + e x^2))`

now grouping

`{(b-1 - 4 x^2 + a (g + f x + e x^2)=0),(d - 12 x^2 + c (g + f x + e x^2)=0):}`

we get the conditions

`{(b + a g =1), (a f = 0), (a e = 4), (d + c g= 0), (c f=0), (c e=12):}`

solving we get at

`{(a = 4/e),(b = (e-4g)/e),(c=12/e),(d = -12g/e),(f=0):}`

Here `e, g` are simply parameters. So fixing `e = 1, g = 1` we get at

`{(a=4),(b=-3),(c=12),(d=-12),(e=1),(f=0),(g=1):}`

or

`u(x) = (4 x-3)/(12 x-12)`
`v(x) = x^2+1`