MATH HELP PLEASE!!! PLEASE HELP!!!

The answer is:  [C]:  " f(c) = [tex] \frac{9}{5} [/tex] c  + 32 " .
________________________________________________________

Explanation:
________________________________________________________
Given the original function:  

" c(y) = (5/9) (x − 32) " ; in which "x = f" ; and "y = c(f) " ;
________________________________________________________
→  Write the original function as:  " y = (5/9) (x − 32) " ; 

Now, change the "y" to an "x" ; and the "x" to a "y"; and rewrite; as follows:
________________________________________________________
    x = (5/9) (y − 32) ; 

Now, rewrite THIS equation; by solving for "y" ; in terms of "x" ; 
_____________________________________________________
→ That is, solve this equation for "y" ; with "c" as an "isolated variable" on the
 "left-hand side" of the equation:

We have:

→  x  =  " (  [tex] \frac{5}{9} [/tex]  ) * (y − 32) " ;

Let us simplify the "right-hand side" of the equation:
_____________________________________________________

Note the "distributive property" of multiplication:
__________________________________________
a(b + c) = ab + ac ;  AND:

a(b – c) = ab – ac .
__________________________________________

As such:
__________________________________________

" ([tex] \frac{5}{9} [/tex]) * (y − 32) " ; 

=  [ ([tex] \frac{5}{9} [/tex]) * y ]   −  [ ([tex] \frac{5}{9} [/tex]) * (32) ] ; 


=  [ ([tex] \frac{5}{9} [/tex]) y ]  − [ ([tex] \frac{5}{9} [/tex]) * ([tex] \frac{32}{1}) [/tex]" ;

=  [ ([tex] \frac{5}{9} [/tex]) y ]  − [ ([tex] \frac{(5*32)}{(9*1)} [/tex] ] ; 

=  [ ([tex] \frac{5}{9} [/tex]) y ]  −  [ ([tex] \frac{(160)}{(9)} [/tex] ] ; 

= [ ([tex] \frac{5y}{9} [/tex]) ]  −  [ ([tex] \frac{(160)}{(9)} [/tex] ] ; 

= [ [tex] \frac{(5y-160)}{9} [/tex] ] ;  
_______________________________________________
And rewrite as:  

→  " x  =  [tex] \frac{(5y-160)}{9} [/tex] "  ;

We want to rewrite this; solving for "y";  with "y" isolated as a "single variable" on the "left-hand side" of the equation ;

We have:

→  " x  =  [tex] \frac{(5y-160)}{9} [/tex] "  ; 

↔  " [tex] \frac{(5y-160)}{9} [/tex] = x ; 

Multiply both sides of the equation by "9" ; 

 9 * [tex] \frac{(5y-160)}{9} [/tex]  =  x * 9 ; 

to get:

→  5y − 160 = 9x ; 

Now, add "160" to each side of the equation; as follows:
_______________________________________________________

→  5y − 160 + 160 = 9x + 160 ; 

to get:

→  5y  =  9x + 160 ; 

Now,  divided Each side of the equation by "5" ; 
      to isolate "y" on one side of the equation; & to solve for "y" ; 

→  5y / 5  = (9y + 160) / 5 ; 

to get: 
 
→  y = (9/5)x + (160/5) ; 

→  y =  (9/5)x + 32 ; 

 →  Now, remember we had substituted:  "y" for "c(f)" ; 

Now that we have the "equation for the inverse" ;
     →  which is:  " (9/5)x  + 32" ; 

Remember that for the original ("non-inverse" equation);  "y" was used in place of "c(f)" .  We have the "inverse equation";  so we can denote this "inverse function" ; that is, the "inverse" of "c(f)" as:  "f(c)" .

Note that "x = c" ; 
_____________________________________________________
So, the inverse function is: "  f(c) = (9/5) c  + 32 " .
_____________________________________________________

 The answer is:  " f(c) = [tex] \frac{9}{5} [/tex] c  + 32 " ;
_____________________________________________________
 →  which is:  

→  Answer choice:  [C]:  " f(c) = [tex] \frac{9}{5} [/tex] c  + 32 " .
_____________________________________________________