Question

If f(x) =x by 1-x
and g(x) =x-1 by x

then g of(x) is
​

Answer 1

GIVEN :

The functions f is defined by f(x)=\frac{1}{1-x}f(x)=

1−x

1

and g is defined by g(x)=\frac{x-1}{x}g(x)=

x

x−1

TO FIND :

The value of the composite function (f\circ g)(x)(f∘g)(x)

SOLUTION :

The composite function of f and g is defined by (f\circ g)(x)(f∘g)(x)

(f\circ g)(x)=f(g(x))(f∘g)(x)=f(g(x))

Since g(x)=\frac{x-1}{x}g(x)=

x

x−1

=f(\frac{x-1}{x})=f(

x

x−1

)

Since f(x)=\frac{1}{1-x}f(x)=

1−x

1

put the value of x is \frac{x-1}{x}

x

x−1

in the function f(x)

=\frac{1}{1-(\frac{x-1}{x})}=

1−(

x

x−1

)

1

=\frac{1}{\frac{x-(x-1)}{x}}=

x

x−(x−1)

1

Using the distributive property :

a(x+y)=ax+ay

=\frac{1}{\frac{x-1(x)-1(-1)}{x}}=

x

x−1(x)−1(−1)

1

Adding the like terms

=\frac{1}{\frac{x-x+1}{x}}=

x

x−x+1

1

=\frac{1}{\frac{1}{x}}=

x

1

1

=x=x

∴ (f\circ g)(x)=x(f∘g)(x)=x

Therefore the value of the composite function of f and g (f\circ g)(x)(f∘g)(x) is x

∴ (f\circ g)(x)(f∘g)(x) is x