Insert the composite functions f o g and g o f, when g = x^3 + 1 and f = x^3 + 1.
Understand the Problem
The question asks for the composite functions fg and gf given two functions, g and f. We need to find the expressions for these compositions based on the provided definitions of g and f.
Answer
- \( f \circ g = x^6 + 2x^3 + 2 \) - \( g \circ f = x^6 + 3x^4 + 3x^2 + 2 \)
Answer for screen readers
The composite functions are:
- ( f \circ g = x^6 + 2x^3 + 2 )
- ( g \circ f = x^6 + 3x^4 + 3x^2 + 2 )
Steps to Solve
-
Identify the functions
We have the functions given as follows:
$$ g(x) = x^3 + 1 $$
$$ f(x) = x^2 + 1 $$ -
Calculate ( f \circ g )
To find ( f(g(x)) ), substitute ( g(x) ) into ( f(x) ):
$$ f(g(x)) = f(x^3 + 1) $$
Now substitute ( x^3 + 1 ) into ( f ):
$$ f(g(x)) = (x^3 + 1)^2 + 1 $$ -
Expand ( f(g(x)) )
Expand the expression:
$$ (x^3 + 1)^2 = x^6 + 2x^3 + 1 $$
So,
$$ f(g(x)) = x^6 + 2x^3 + 1 + 1 = x^6 + 2x^3 + 2 $$ -
Calculate ( g \circ f )
Next, find ( g(f(x)) ) by substituting ( f(x) ) into ( g(x) ):
$$ g(f(x)) = g(x^2 + 1) $$
Substituting ( x^2 + 1 ) into ( g ):
$$ g(f(x)) = (x^2 + 1)^3 + 1 $$ -
Expand ( g(f(x)) )
Expand the expression:
$$ (x^2 + 1)^3 = x^6 + 3x^4 + 3x^2 + 1 $$
So,
$$ g(f(x)) = x^6 + 3x^4 + 3x^2 + 1 + 1 = x^6 + 3x^4 + 3x^2 + 2 $$
The composite functions are:
- ( f \circ g = x^6 + 2x^3 + 2 )
- ( g \circ f = x^6 + 3x^4 + 3x^2 + 2 )
More Information
Composite functions combine two functions by substituting one into the other. The outputs of ( g ) are used as inputs for ( f ), and vice versa. This process is fundamental in algebra and calculus.
Tips
- Forgetting to properly substitute the entire function ( g(x) ) into ( f(x) ) or vice versa.
- Not correctly expanding polynomial expressions can lead to incorrect answers.
