1 The meaning of a function of a function

When we use a function like sin 2 x or e ln x or x 2 + 1 we are in fact dealing with a composite function or function of a function .

sin 2 x is the sine function of 2 x . This is, in fact, how we ‘read’ it:

sin 2 x is read ‘sine of 2 x

Similarly e ln x is the exponential function of the logarithm of x :

e ln x is read ‘ e to the power of ln x

Finally x 2 + 1 is also a composite function. It is the square root function of the polynomial x 2 + 1 :

x 2 + 1 is read as the ‘square root of ( x 2 + 1 )

When we talk about a function of a function in a general setting we will use the notation f ( g ( x ) ) where both f and g are functions.

Example 11

Specify the functions f , g for the composite functions

  1. sin 2 x
  2. x 2 + 1
  3. e ln x
Solution
  1. Here f is the sine function and g is the polynomial 2 x . We often write:

    f ( g ) = sin g and g ( x ) = 2 x

  2. Here f ( g ) = g and g ( x ) = x 2 + 1
  3. Here f ( g ) = e g and g ( x ) = ln x

In each case the original function of x is obtained when g ( x ) is substituted into f ( g ) .

Task!

Specify the functions f , g for the composite functions  

  1. cos ( 3 x 2 1 )
  2. sinh ( e x )
  3. ( x 2 + 3 x 1 ) 1 3

f ( g ) = cos g g ( x ) = 3 x 2 1

f ( g ) = sinh g g ( x ) = e x

f ( g ) = g 1 3 g ( x ) = x 2 + 3 x 1