Use the definition of inverses

Question

Use the definition of inverses to determine whether f(x)=(5x-3)/(2x+1) and g(x)=(x+3)/(5-2x) are inverses

Answer 1

f(g(x))

= (5g(x)-3)/(2g(x)+1)

= [5(x+3)-3(5-2x)]/[2(x+3)+(5-2x)], after multiplying by (5-2x)/(5-2x)

= 11x/11

= x

 

g(f(x))

= (f(x)+3)/(5-2f(x))

= [(5x-3)+3(2x+1)]/[(5(2x+1)-2(5x-3)], after multiplying by (2x+1)/(2x+1)

= 11x/11

= x

 

So, f(x) and g(x) are inverses.