Concatenating functions

If for x∈Dg a function z=g(x) with the value range Wg is given and moreover for z∈Wg a function y=f(z), then y=f(g(x)) is called (with x ∈Dg) is called an indirect (concatenated) function of x.
Notation: y=f∘g (read: f "roller" g or f "squiggle" g).

Notes:  The concatenation requirement Wg⊆Df must be observed.

f∘g means: First apply g then f (ie f after g).

The function f is called the outer function, the function g the inner function of the concatenated function y=f(g(x)).

The concatenation f(g(x)) is defined for all x for which the function values ​​of g (ie g(x)) belong to the domain of definition of f -  ultimate maths solver  . A chaining of functions is only possible if the intersection of the definition range of the outer function and the value range of the inner function is not empty.

It is possible to form a concatenation of more than two functions. For example, the triple concatenation f∘g∘h consists of applying first h, then g and then f, taking into account the preconditions:

 f∘g∘h=f(g(h(x))).

You can proceed in the same way with any number of functions (again taking into account the prerequisites).

Example:  The two possible concatenations of the following functions are to be determined:

f: [0; 8]→R, f(x)=5x-10g: R0→R0, g(x)=√x

1st part of the solution:

 f∘g=f(g(x))

It is Wg=R0⊆[0; 8]=Df, but Wg∩Df=R0∩[0; 8]=[0; 8]≠0; ie, we must first restrict the domain of definition of g such that g(x)∈[0; 8], ie,

0≤g(x)≤8⇔0≤√x≤8⇔x≤64.

Then we obtain:

 f∘g=f(g(x))=5√x---10, x∈[0; 64]

2nd part of the solution:

 g∘f=g(f(x))

It is Wf=R⊆R0=Dg, but Wf∩Dg=R∩R0=R0≠0; ie, we must first restrict the domain of definition [0; 8] of f such that f(x)∈R0, ie, f(x)≥0⇔5x-10≥0⇔5x≥10⇔x≥2 and x≤8.

Then we obtain:

 g∘f=g(f(x))=√(5x-10), x∈[2; 8th].

Important:  It is f∘g≠g∘f, ie, when concatenating functions, the order must be observed.