Answer :
Sure! Let's find two functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] such that their composition gives us the given functions [tex]\( h(x) \)[/tex].
### Problem 71: [tex]\( h(x) = (2x + 1)^2 \)[/tex]
We want to find two functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] such that:
[tex]\[
(f \circ g)(x) = h(x) = (2x + 1)^2
\][/tex]
To do this, we can choose:
1. Function [tex]\( g(x) \)[/tex]:
- Let [tex]\( g(x) = 2x + 1 \)[/tex].
- This function modifies the input [tex]\( x \)[/tex] to become [tex]\( 2x + 1 \)[/tex].
2. Function [tex]\( f(x) \)[/tex]:
- Let [tex]\( f(x) = x^2 \)[/tex].
- This function takes an input [tex]\( x \)[/tex] and squares it.
By composing these functions, we have:
[tex]\[
(f \circ g)(x) = f(g(x)) = f(2x + 1) = (2x + 1)^2
\][/tex]
This matches our given [tex]\( h(x) \)[/tex], so:
- [tex]\( f(x) = x^2 \)[/tex]
- [tex]\( g(x) = 2x + 1 \)[/tex]
### Problem 72: [tex]\( h(x) = (1-x)^3 \)[/tex]
For this problem, we need functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] such that:
[tex]\[
(f \circ g)(x) = h(x) = (1-x)^3
\][/tex]
We can choose:
1. Function [tex]\( g(x) \)[/tex]:
- Let [tex]\( g(x) = 1 - x \)[/tex].
- This function changes the input [tex]\( x \)[/tex] to [tex]\( 1 - x \)[/tex].
2. Function [tex]\( f(x) \)[/tex]:
- Let [tex]\( f(x) = x^3 \)[/tex].
- This function cubes the input [tex]\( x \)[/tex].
By composing these functions, we have:
[tex]\[
(f \circ g)(x) = f(g(x)) = f(1-x) = (1-x)^3
\][/tex]
This matches the given [tex]\( h(x) \)[/tex], so:
- [tex]\( f(x) = x^3 \)[/tex]
- [tex]\( g(x) = 1 - x \)[/tex]
These are the functions that, when composed, will create the desired composite functions for each problem!
### Problem 71: [tex]\( h(x) = (2x + 1)^2 \)[/tex]
We want to find two functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] such that:
[tex]\[
(f \circ g)(x) = h(x) = (2x + 1)^2
\][/tex]
To do this, we can choose:
1. Function [tex]\( g(x) \)[/tex]:
- Let [tex]\( g(x) = 2x + 1 \)[/tex].
- This function modifies the input [tex]\( x \)[/tex] to become [tex]\( 2x + 1 \)[/tex].
2. Function [tex]\( f(x) \)[/tex]:
- Let [tex]\( f(x) = x^2 \)[/tex].
- This function takes an input [tex]\( x \)[/tex] and squares it.
By composing these functions, we have:
[tex]\[
(f \circ g)(x) = f(g(x)) = f(2x + 1) = (2x + 1)^2
\][/tex]
This matches our given [tex]\( h(x) \)[/tex], so:
- [tex]\( f(x) = x^2 \)[/tex]
- [tex]\( g(x) = 2x + 1 \)[/tex]
### Problem 72: [tex]\( h(x) = (1-x)^3 \)[/tex]
For this problem, we need functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] such that:
[tex]\[
(f \circ g)(x) = h(x) = (1-x)^3
\][/tex]
We can choose:
1. Function [tex]\( g(x) \)[/tex]:
- Let [tex]\( g(x) = 1 - x \)[/tex].
- This function changes the input [tex]\( x \)[/tex] to [tex]\( 1 - x \)[/tex].
2. Function [tex]\( f(x) \)[/tex]:
- Let [tex]\( f(x) = x^3 \)[/tex].
- This function cubes the input [tex]\( x \)[/tex].
By composing these functions, we have:
[tex]\[
(f \circ g)(x) = f(g(x)) = f(1-x) = (1-x)^3
\][/tex]
This matches the given [tex]\( h(x) \)[/tex], so:
- [tex]\( f(x) = x^3 \)[/tex]
- [tex]\( g(x) = 1 - x \)[/tex]
These are the functions that, when composed, will create the desired composite functions for each problem!
