Answer :
Answer
Find out f(f(x)) .
To prove
As given in the question
f(x) = x² + 1
Now
f(f(x)) = f (x² + 1)
Now find out the value of f (x² + 1).
f(f(x)) = f (x² + 1)
= (x² + 1)² + 1
Using formula
(a + b)² = a² + b² + 2ab
Apply in the above function
f(f(x)) = (x² + 1)² + 1
[tex]f(f(x)) = x^{4} +1 +2x^{2}+ 1[/tex]
Simplify
[tex]f(f(x)) = x^{4} +2x^{2}+ 2[/tex]
Thus the value of f(f(x)) is [tex]f(f(x)) = x^{4} +2x^{2}+ 2[/tex] .
You can use the given function's definition and put that as input in the given function to evaluate function of function.
The expression for f(f(x)) is
[tex]f(f(x)) = x^4 + 2x^2 + 2[/tex]
How to evaluate function of functions?
We pass the output of the first function as input to the second function.
Suppose the given functions are
[tex]f(x)[/tex] and [tex]g(x)[/tex]
Then the function of functions for "g" taking input as "f" is written as
[tex]g(f(x)) = g \circ f(x)\\f(g(x)) = f \circ g(x)[/tex]
This above is the notation we use to write composite functions.
Thus, as the given function is
[tex]f(x) = x^2 + 1[/tex]
So, the function of function would be:
[tex]f(f(x)) = (f(x))^2 + 1 = (x^2 + 1)^2 + 1 = x^4 + 2x^2 + 1 + 1 \\\\f(f(x)) = x^4 + 2x^2 + 2[/tex]
The expression for f(f(x)) is
[tex]f(f(x)) = x^4 + 2x^2 + 2[/tex]
Learn more about composite functions here:
https://brainly.com/question/24780056