Composite Function Problem

Question

 Given the composite function h(x) = f(f(x)) = (x+1)2+1h(x) = f(f(x)) = (x+1)2+1 and f(x) = x+1f(x) = x+1.

Find the value of h(3)h(3).

Answer 1

We want to find h(3), given h(x) = f( f(x) ), where f(x) = x + 1

[ f(x) = x + 1 is quite hard to make out in the original problem description with all the repeated text. ]

Before we do any work, let’s briefly note what we’re doing here… we are taking the “output” of f(x), and feeding that output back as “input” into itself. There is nothing wrong with this, as long as it is clear that anything in the “range” of f(x) is a suitable element in the “domain” of f(x).

In this case, f(x) may have as its “domain” anything in the set of real numbers, denoted ℝ. Luckily, the range of f(x) is also in ℝ, so the "output" of f() can be fed as ‘input’ back into itself.

This is not always the case, so some care must be exercised… suppose f(x) = -√x + 1, then does it make sense to speak of f( f(x) ) ? f(4) = -2 + 1 = -1, then what is f( -1 ) = -√(-1) + 1 ? Whoops ! But f( f(1/4) ) is fine, as would be f( f(1) ).

Just a reminder that y = f(x) is not always just a "box" that cranks on things, and spits out answers. f() is a "relation" between elements in a Domain and other elements in a Range. Sometimes Domains and Ranges can be the same, or coexist peacefully. When the Domain is less flexible, like Domain = ℝ+ instead of ℝ, we need to be more careful.

We have: h(x) = f( f(x) ), f(x) = x + 1 ==> h(3) = f( f(3) )

f(3) = (3 + 1) = 4 ==> h(3) = f( f(3) ) = f( 4 ) = 4 + 1 = 5

More generally h(x) = f( f(x) ) = f(x) + 1 = (x + 1) + 1 ==> h(x) = x + 2, which confirms the result f(3) = 5 above.

[NB. I do not see why h(x) would become (x+1)² + 1, as hinted in the original problem description, but the description is a bit messy ]

In later branches of Mathematics, a Composite Function can be indicated using either h(x) = f( g(x) ), or h(x) = (f∘g)(x)... that little "o" is not a normal little "o", but the "ring operator”. Unlike a little “o”, it is not on the baseline, but a bit higher up. Fwiw. It is read "of", as in: "h of x = f of g of x”, and is used to not confuse it with (f*g)(x) = f(x)*g(x). fog(x) could be read as “I really don’t know”.

Answer 2

Given that f(x) = x+1.

and composite function h(x) = f(f(x)) = (x+1)² +1.

if x=3: Then h(x)= h(3) = f(f(3)) = (3 + 1)^2 + 1

h(3)=(4)²+1

= 17.

( In the above problem, the composite function h(x) is explicitly given. Now you need to think, if the question was given as:

The h(x) = x^2+1, and f(x) = x+1. Then how you evaluate h(f(x)). )

Answer 3

There are two ways to approach this:

Method 1: Directly using the given composite function form

Since we are explicitly given h(x)=(x+1)2+1h(x)=(x+1)2+1, we can directly substitute x=3x=3 into this expression.

h(3)=(3+1)2+1h(3)=(3+1)2+1 h(3)=(4)2+1h(3)=(4)2+1 h(3)=16+1h(3)=16+1 h(3)=17h(3)=17

Answer 4

There are two ways to approach this:

Method 1: Directly using the given composite function form

Since we are explicitly given h(x)=(x+1)2+1h(x)=(x+1)2+1, we can directly substitute x=3x=3 into this expression.

h(3)=(3+1)2+1h(3)=(3+1)2+1 h(3)=(4)2+1h(3)=(4)2+1 h(3)=16+1h(3)=16+1 h(3)=17h(3)=17