Westonci.ca connects you with experts who provide insightful answers to your questions. Join us today and start learning! Discover solutions to your questions from experienced professionals across multiple fields on our comprehensive Q&A platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
To verify if [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex], we need to check if applying [tex]\( g \)[/tex] to [tex]\( f(x) \)[/tex] yields the original input, [tex]\( x \)[/tex]. In mathematical terms, we need to find out if [tex]\( g(f(x)) = x \)[/tex].
Given:
[tex]\[ f(x) = 5x - 25 \][/tex]
[tex]\[ g(x) = \frac{1}{5} x + 5 \][/tex]
Let's determine [tex]\( g(f(x)) \)[/tex]:
1. First, apply [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = 5x - 25 \][/tex]
2. Now apply [tex]\( g \)[/tex] to [tex]\( f(x) \)[/tex]:
[tex]\[ g(f(x)) = g(5x - 25) \][/tex]
Substitute [tex]\( 5x - 25 \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(5x - 25) = \frac{1}{5} (5x - 25) + 5 \][/tex]
Simplify the expression step-by-step:
[tex]\[ g(5x - 25) = \frac{1}{5} \cdot 5x - \frac{1}{5} \cdot 25 + 5 \][/tex]
[tex]\[ g(5x - 25) = x - 5 + 5 \][/tex]
[tex]\[ g(5x - 25) = x \][/tex]
We have shown that [tex]\( g(f(x)) = x \)[/tex]. Therefore, [tex]\( g(x) \)[/tex] is indeed the inverse of [tex]\( f(x) \)[/tex].
The correct expression to verify that [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex] is:
[tex]\[ \frac{1}{5}(5x - 25) + 5 \][/tex]
So, the correct answer is:
[tex]\[ \frac{1}{5}(5x - 25) + 5 \][/tex]
Given:
[tex]\[ f(x) = 5x - 25 \][/tex]
[tex]\[ g(x) = \frac{1}{5} x + 5 \][/tex]
Let's determine [tex]\( g(f(x)) \)[/tex]:
1. First, apply [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = 5x - 25 \][/tex]
2. Now apply [tex]\( g \)[/tex] to [tex]\( f(x) \)[/tex]:
[tex]\[ g(f(x)) = g(5x - 25) \][/tex]
Substitute [tex]\( 5x - 25 \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(5x - 25) = \frac{1}{5} (5x - 25) + 5 \][/tex]
Simplify the expression step-by-step:
[tex]\[ g(5x - 25) = \frac{1}{5} \cdot 5x - \frac{1}{5} \cdot 25 + 5 \][/tex]
[tex]\[ g(5x - 25) = x - 5 + 5 \][/tex]
[tex]\[ g(5x - 25) = x \][/tex]
We have shown that [tex]\( g(f(x)) = x \)[/tex]. Therefore, [tex]\( g(x) \)[/tex] is indeed the inverse of [tex]\( f(x) \)[/tex].
The correct expression to verify that [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex] is:
[tex]\[ \frac{1}{5}(5x - 25) + 5 \][/tex]
So, the correct answer is:
[tex]\[ \frac{1}{5}(5x - 25) + 5 \][/tex]
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.