Question

If
the function f: R->R is given
by f(x) = x²+2 and g: R- R be given
by g(x)=x/x-1
Find fog
gof
fog and
x-1​

Answer 1

Answer :

• fog(x) = (3x² - 4x + 2) / (x² - 2x + 1)

• gof(x) = (x² + 2)/(x² + 1)

• fof(x) = x⁴ + 4x² + 6

Solution :

★ Given functions :

• f : R → R , f(x) = x² + 2

• g : R - {1} → R , g(x) = x/(x - 1)

★ To find :

• fog(x) = ?

• gof(x) = ?

• fof(x) = ?

★ fog(x) = f [ g(x) ]

=> fog(x) = f(x/x-1)

=> fog(x) = (x/x-1)² + 2

=> fog(x) = x²/(x-1)² + 2

=> fog(x) = x²/(x² - 2x + 1) + 2

=> fog(x) = [ x² + 2(x² - 2x + 1) ] / (x² - 2x + 1)

=> fog(x) = (x² + 2x² - 4x + 2) / (x² - 2x + 1)

=> fog(x) = (3x² - 4x + 2) / (x² - 2x + 1)

★ gof(x) = g [ f(x) ]

=> gof(x) = g(x² + 2)

=> gof(x) = (x² + 2) / [ (x² + 2) - 1 ]

=> gof(x) = (x² + 2) / (x² + 2 - 1)

=> gof(x) = (x² + 2)/(x² + 1)

★ fof(x) = f [ f(x) ]

=> fof(x) = f(x² + 2)

=> fof(x) = (x² + 2)² + 2

=> fof(x) = (x²)² + 2•x²•2 + 2² + 2

=> fof(x) = x⁴ + 4x² + 4 + 2

=> fof(x) = x⁴ + 4x² + 6