To determine whether [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are inverse functions, we need to check whether their compositions simplify to the identity function [tex]\( x \)[/tex]. Specifically, we need to check if:
1. [tex]\( f(g(x)) = x \)[/tex]
2. [tex]\( g(f(x)) = x \)[/tex]
Let's start with [tex]\( f(g(x)) \)[/tex]:
Given:
[tex]\[ f(x) = \sqrt[3]{x-1} \][/tex]
[tex]\[ g(x) = x^3 + 1 \][/tex]
First, find [tex]\( f(g(x)) \)[/tex]:
[tex]\[ f(g(x)) = f(x^3 + 1) \][/tex]
Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(x^3 + 1) = \sqrt[3]{(x^3 + 1) - 1} \][/tex]
[tex]\[ f(x^3 + 1) = \sqrt[3]{x^3} \][/tex]
[tex]\[ f(x^3 + 1) = (\sqrt[3]{x^3}) \][/tex]
So:
[tex]\[ f(g(x)) = (x^3)^{1/3} \][/tex]
Simplifying further, we get:
[tex]\[ f(g(x)) = x \][/tex]
Now, let's check [tex]\( g(f(x)) \)[/tex]:
[tex]\[ g(f(x)) = g(\sqrt[3]{x-1}) \][/tex]
Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(\sqrt[3]{x-1}) = (\sqrt[3]{x-1})^3 + 1 \][/tex]
Simplify:
[tex]\[ g(\sqrt[3]{x-1}) = (x-1) + 1 \][/tex]
[tex]\[ g(\sqrt[3]{x-1}) = x \][/tex]
However, upon closer inspection of the simplified form provided earlier:
[tex]\[ g(\sqrt[3]{x-1}) = (x-1) + 1 \][/tex]
This is numerically equivalent to [tex]\( x \)[/tex], but simplifying further, we get:
[tex]\[ g(f(x)) = x \][/tex]
Based on our findings:
1. [tex]\( f(g(x)) = (x^3)^{1/3} \)[/tex], which simplifies to [tex]\( x \)[/tex]
2. [tex]\( g(f(x)) = (\sqrt[3]{x-1})^3 + 1 \)[/tex], which simplifies to [tex]\( x \)[/tex]
In conclusion, because both [tex]\( f(g(x)) \)[/tex] and [tex]\( g(f(x)) \)[/tex] simplify to [tex]\( x \)[/tex], we can conclude that [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are NOT inverse functions.
