inverse composition

Question

show that f(x)=x+5/4 and g(x)=4x-5 are inverses using composition. 

Answer 1

The f(x) and g(x) stated in the problem are not inverses.  However, if you restate f(x) with parentheses, then they are inverses.

 

f(x) = (x + 5)/4  [Note the parentheses.  This is NOT the same as x + 5/4]

 

g(x) = 4x - 5

 

If (fοg)(x) = (gοf)(x) = x, then f and g are inverses.

 

(fοg)(x) = [g(x) + 5]/4 = [(4x - 5) + 5]/4 = x

 

(gοf)(x) = 4f(x) - 5 = 4[(x + 5)/4] - 5 = x

 

Hence, f = g^-1 and g = f^-1