If f(x) =x by 1-x
and g(x) =x-1 by x
then g of(x) is
Answers
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