How do I view the list of functions a Linux shared library is exporting?

On a MAC, you need to use nm *.o | c++filt, as there is no -C option in nm.

What you need is nm and its -D option:

$ nm -D /usr/lib/libopenal.so.1
.
.
.
00012ea0 T alcSetThreadContext
000140f0 T alcSuspendContext
         U atanf
         U calloc
.
.
.

Exported sumbols are indicated by a T. Required symbols that must be loaded from other shared objects have a U. Note that the symbol table does not include just functions, but exported variables as well.

See the nm manual page for more information.

Among other already mentioned tools you can use also readelf (manual). It is similar to objdump but goes more into detail. See this for the difference explanation.

$ readelf -sW /lib/liblzma.so.5 |head -n10

Symbol table '.dynsym' contains 128 entries:
   Num:    Value  Size Type    Bind   Vis      Ndx Name
     0: 00000000     0 NOTYPE  LOCAL  DEFAULT  UND
     1: 00000000     0 FUNC    GLOBAL DEFAULT  UND pthread_mutex_unlock@GLIBC_2.0 (4)
     2: 00000000     0 FUNC    GLOBAL DEFAULT  UND pthread_mutex_destroy@GLIBC_2.0 (4)
     3: 00000000     0 NOTYPE  WEAK   DEFAULT  UND _ITM_deregisterTMCloneTable
     4: 00000000     0 FUNC    GLOBAL DEFAULT  UND memmove@GLIBC_2.0 (5)
     5: 00000000     0 FUNC    GLOBAL DEFAULT  UND free@GLIBC_2.0 (5)
     6: 00000000     0 FUNC    GLOBAL DEFAULT  UND memcpy@GLIBC_2.0 (5)

objdump -T *.so may also do the job