Westonci.ca is your go-to source for answers, with a community ready to provide accurate and timely information. Discover the answers you need from a community of experts ready to help you with their knowledge and experience in various fields. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
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.
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.
We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.