How do AX, AH, AL map onto EAX?

No, that's not quite right.

EAX is the full 32-bit value
AX is the lower 16-bits
AL is the lower 8 bits
AH is the bits 8 through 15 (zero-based)

So AX is composed of AH:AL halves, and is itself the low half of EAX. (The upper half of EAX isn't directly accessible as a 16-bit register; you can shift or rotate EAX if you want to get at it.)

For completeness, in addition to the above, which was based on a 32-bit CPU, 64-bit Intel/AMD CPUs have

RAX, which hold a 64-bit value, and where EAX is mapped to the lower 32 bits.

All of this also applies to EBX/RBX, ECX/RCX, and EDX/RDX. The other registers like EDI/RDI have a DI low 16-bit partial register, but no high-8 part, and the low-8 DIL is only accessible in 64-bit mode: Assembly registers in 64-bit architecture

Writing AL, AH, or AX leaves other bytes unmodified in the full AX/EAX/RAX, for historical reasons. i.e. it has to merge a new AL into the full RAX, for example. (In 32 or 64-bit code, prefer a movzx eax, byte [mem] or movzx eax, word [mem] load if you don't specifically want this merging: Why doesn't GCC use partial registers?)

Writing EAX zero-extends into RAX. (Why do x86-64 instructions on 32-bit registers zero the upper part of the full 64-bit register?)

Again, all of this applies to every register, not just RAX. e.g. writing DI or DIL merges into the old RDI, writing EDI zero-extends and overwrites the full RDI. Same for R10B or R10W writes merging, writing R10D leaving R10 independent of the old R10 value.

AX is the 16 lower bits of EAX. AH is the 8 high bits of AX (i.e. the bits 8-15 of EAX) and AL is the least significant byte (bits 0-7) of EAX as well as AX.

Example (Hexadecimal digits):

EAX: 12 34 56 78
AX: 56 78
AH: 56
AL: 78

| 0000 0001 0010 0011 0100 0101 0110 0111 | ------> EAX

|                     0100 0101 0110 0111 | ------> AX

|                               0110 0111 | ------> AL

|                     0100 0101           | ------> AH

no your ans is Wrong

Selection of Al and Ah is from AX not from EAX

e.g

EAX=0000 0000 0000 0000 0000 0000 0000 0111

So if we call AX it should return

0000 0000 0000 0111

if we call AH it should return

0000 0000

and when we call AL it should return

0000 0111

Example number 2

EAX: 22 33 55 77
AX: 55 77
AH: 55    
AL: 77

example 3

EAX: 1111 0000 0000 0000 0000 0000 0000 0111    
AX= 0000 0000 0000 0111
AH= 0000 0000
AL= 0000 0111  

No -- AL is the 8 least significant bits of AX. AX is the 16 least significant bits of EAX.

Perhaps it's easiest to deal with if we start with 04030201h in eax. In this case, AX will contain 0201h, AH wil contain 02h and AL will contain 01h.