4 Find the inverse of functions a f R R fx 5x 38 b f R 5

4) Find the inverse of functions
a) f: R -> R, f(x) = (5x + 3)/8;
b) f: R - {-5/2} -> R - {3/2}, f(x) = 3x/(2x + 5);
c) f: R+ -> (0,1), f(x) = 1/(x + 1);

Solution

a) f(x)=(5x+3)/8

y=(5x+3)/8

interchange x and y

x=(5y+3)/8

8x-3=5y

(8x-3)/5=y

f^-1(x)=(8x-3)/5

b) y=3x/(2x+5)

x=3y/(2y+5)

x(2y+5)=3y

2xy+5x=3y

5x=3y-2xy

5x=y(3-2x)

y=5x/(3-2x)

  f^-1(x)=5x/(3-2x) for R-{3/2} ->R-{-5/2}

C) y=1/(x+1)

x=1/(y+1)

xy+x=1

xy=1-x

y=(1-x)/x

  f^-1(x)=(1-x)/x