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Let *f* : A â†’ B and *g* : B â†’ C be two functions. Then the composition of *f* and *g*, denoted by *gof*, is defined as function *gof* : A â†’ C given by

*gof* (*x*) = *g*(*f* (*x*)), âˆ€ x âˆˆ A.

Points to remember:

(i) If f:A â†’ B and g:B â†’ C, then gof:A â†’ C.

(ii) Function gof is possible only if the range of function f is the domain of function g.

(iii) When gof is possible, fog may or may not be possible.

(iv) If both gof and fog are possible, then they may or may not be equal.

Ex:

Suppose you are at a furniture shop and the shopkeeper giving a discount of 10 percent on the purchase of furniture exceeding 5000 rupees.

Sol:

Let us now to calculate the discount if a purchase of x rupees is made.

Amount over Rs 5000= x â€“ 5000

Let g: R â†’ R such that g(x) = x â€“ 5000

Discount = 10 % of (x â€“ 5000)

Let f: R â†’ R such that f(x) = 10% of (x â€“ 5000)

Let x = 8000

fog(8000) = f(g(8000)) ('.' g(x) = x â€“ 5000)

= f(8000â€“5000)

=f(3000) = 10 % of 3000 = 300 ( f(x) = 10% of (x â€“ 5000) ).

Properties of Composite Functions

Let f: R â†’ R and g: R â†’ R defined by f(x) = 2x and g(x) = 3x^{2}.

**To find**: gof and fog

gof(x) = g[f(x)]

= g[2x] ['.' f(x) = 2x]

= 3(2x)^{2 } ['.' g(x) = 3x^{2}]

gof(x) = 12x^{2}

fog(x) = f[g(x)]

= f[3x^{2}] [Since, g(x) = 3x^{2}]

= 2(3x^{2}) [ '.' f(x) = 2x]

fog(x) = 6x^{2}

Observe that gof(x) â‰ fog(x)

Hence, we can say that the two functions gof and fog are unequal.

The composition of functions does not satisfy the commutative property.

Let *f* : A â†’ B and *g* : B â†’ C be two functions. Then the composition of *f* and *g*, denoted by *gof*, is defined as function *gof* : A â†’ C given by

*gof* (*x*) = *g*(*f* (*x*)), âˆ€ x âˆˆ A.

Points to remember:

(i) If f:A â†’ B and g:B â†’ C, then gof:A â†’ C.

(ii) Function gof is possible only if the range of function f is the domain of function g.

(iii) When gof is possible, fog may or may not be possible.

(iv) If both gof and fog are possible, then they may or may not be equal.

Ex:

Suppose you are at a furniture shop and the shopkeeper giving a discount of 10 percent on the purchase of furniture exceeding 5000 rupees.

Sol:

Let us now to calculate the discount if a purchase of x rupees is made.

Amount over Rs 5000= x â€“ 5000

Let g: R â†’ R such that g(x) = x â€“ 5000

Discount = 10 % of (x â€“ 5000)

Let f: R â†’ R such that f(x) = 10% of (x â€“ 5000)

Let x = 8000

fog(8000) = f(g(8000)) ('.' g(x) = x â€“ 5000)

= f(8000â€“5000)

=f(3000) = 10 % of 3000 = 300 ( f(x) = 10% of (x â€“ 5000) ).

Properties of Composite Functions

Let f: R â†’ R and g: R â†’ R defined by f(x) = 2x and g(x) = 3x^{2}.

**To find**: gof and fog

gof(x) = g[f(x)]

= g[2x] ['.' f(x) = 2x]

= 3(2x)^{2 } ['.' g(x) = 3x^{2}]

gof(x) = 12x^{2}

fog(x) = f[g(x)]

= f[3x^{2}] [Since, g(x) = 3x^{2}]

= 2(3x^{2}) [ '.' f(x) = 2x]

fog(x) = 6x^{2}

Observe that gof(x) â‰ fog(x)

Hence, we can say that the two functions gof and fog are unequal.

The composition of functions does not satisfy the commutative property.