Composite Function Solver

In mathematics, composite function is the application of one function to the results of another. For instance, the functions f: X → Y and g: Y → Z can be composed by computing the output of g when it has an argument of f(x) instead of x. intuitively, if z is a function g of y and y is a function f of x, and then z is a function of x                                  .( Source: Wikipedia )

In the following section we are going to see properties of composite function and some problems on composite function by using composite function solver.


Composite function solver

Fig(i) Composite function

Properties of composite function solver:

The composite function solver has the following properties.

The notation for indicate the composite function is  `@` .That is (f ∘ g).Where f and g are two functions. We can also say that f `@` g as f circle g or g composed with f, g after f , g of f.

f ∘ (g ∘ h) = (f ∘ g) ∘ h

(f ∘ g)(x) =f(g(x))

If the function f and g are commute each other ,we can define (f ∘ g) = (g ∘ f)

(f ∘ f)(x) =f 2 ( x )

(f ∘ f ∘ f )(x) = f 3 (x)


Problems on composite function solver :

Problem 1:

f(x) = 2x+5 and g(x) = 4x. Find f(g(x)).

Solution:

Given, f(x) = 2x+5

g(x) = 4x

We need to find f(g(x)) .

That is in the function f(x) ,substitute x =g(x) = 4x

f(x) = 2x+5

f(g(x)) =  2(4x) + 5

= 8x + 5

f(g(x)) = 8x + 5

Answer: 8x + 5

Problem 2:

f(x) = 3x2 + 5x -6 and g(x) = x -5 .Find (f ∘ g)(x)

Solution:

Given,  f(x) = 3x2 + 5x -6

g(x) = x -5

We need to find (f ∘ g)(x),

According to composite function solver property , (f ∘ g)(x) = f(g(x))

To obtain  f(g(x)) ,Substitute x =g(x) in f(x),

f(x) = 3x2 + 5x -6

f(g(x)) = 3( x -5 )2 + 5 ( x -5 ) -6

= 3 ( x2 - 10x +25) + 5x -25 - 6

=3x2 - 30x +75 + 5x -31

= 3x2 - 25 x +44

Answer: f(g(x)) = 3x2 - 25 x +44


Problem 3:

h(x) = x+2 and g(x) = -5x .Find h(g(6))

Solution:

Given , h(x) = x+2

g(x) = -5x

g( 6 ) = -5(6)

= -30

f(g(6)) = x+2

= -30 + 2

= -28

Answer: f(g(6)) = -28