Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
To determine which expression could be used to verify if [tex]\( g(x) = \frac{1}{5} x + 5 \)[/tex] is the inverse of [tex]\( f(x) = 5x - 25 \)[/tex], we need to follow these steps:
1. Understand the Definitions:
- The function [tex]\( f(x) = 5x - 25 \)[/tex].
- The function [tex]\( g(x) = \frac{1}{5} x + 5 \)[/tex].
2. Verify Inverse Relationship:
- For [tex]\( g(x) \)[/tex] to be the inverse of [tex]\( f(x) \)[/tex], we must have:
[tex]\[ g(f(x)) = x \quad \text{and} \quad f(g(x)) = x \][/tex]
3. Calculate [tex]\( g(f(x)) \)[/tex]:
[tex]\[ g(f(x)) = g(5x - 25) \][/tex]
Substitute [tex]\( 5x - 25 \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(5x - 25) = \frac{1}{5} (5x - 25) + 5 = x - 5 + 5 = x \][/tex]
[tex]\( g(f(x)) = x \)[/tex], this part holds true.
4. Calculate [tex]\( f(g(x)) \)[/tex]:
[tex]\[ f(g(x)) = f\left( \frac{1}{5} x + 5 \right) \][/tex]
Substitute [tex]\( \frac{1}{5} x + 5 \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f\left(\frac{1}{5} x + 5\right) = 5\left(\frac{1}{5} x + 5\right) - 25 = x + 25 - 25 = x \][/tex]
[tex]\( f(g(x)) = x \)[/tex], this part also holds true.
5. List Possible Expressions:
Evaluate which of the given expressions simplifies to [tex]\( x \)[/tex]:
- [tex]\(\frac{1}{5}\left(\frac{1}{5} x + 5\right) + 5\)[/tex]
[tex]\[ \frac{1}{5}\left(\frac{1}{5} x + 5\right) + 5 = \frac{1}{25} x + 1 + 5 = \frac{1}{25} x + 6 \][/tex]
This does not simplify to [tex]\( x \)[/tex].
- [tex]\(\frac{1}{5}(5x - 25) + 5\)[/tex]
[tex]\[ \frac{1}{5}(5x - 25) + 5 = x - 5 + 5 = x \][/tex]
This simplifies to [tex]\( x \)[/tex].
- [tex]\(\frac{1}{\left(\frac{1}{5} x + 5\right)}\)[/tex]
[tex]\[ \frac{1}{\left(\frac{1}{5} x + 5\right)} = \text{This simplifies to} \frac{1}{5}x + 5 \text{ if this is the inverse} \][/tex]
This does not simplify to [tex]\( x \)[/tex].
- [tex]\( 5\left(\frac{1}{5} x + 5\right) + 5 \)[/tex]
[tex]\[ 5\left(\frac{1}{5} x + 5\right) + 5 = x + 25 + 5 = x + 30 \][/tex]
This does not simplify to [tex]\( x \)[/tex].
Based on this analysis, the second expression, [tex]\(\frac{1}{5}(5x - 25) + 5 \)[/tex], is the one that can be used to verify if [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex].
Therefore, the correct option is:
[tex]\[ \boxed{\frac{1}{5}(5x - 25) + 5} \][/tex]
1. Understand the Definitions:
- The function [tex]\( f(x) = 5x - 25 \)[/tex].
- The function [tex]\( g(x) = \frac{1}{5} x + 5 \)[/tex].
2. Verify Inverse Relationship:
- For [tex]\( g(x) \)[/tex] to be the inverse of [tex]\( f(x) \)[/tex], we must have:
[tex]\[ g(f(x)) = x \quad \text{and} \quad f(g(x)) = x \][/tex]
3. Calculate [tex]\( g(f(x)) \)[/tex]:
[tex]\[ g(f(x)) = g(5x - 25) \][/tex]
Substitute [tex]\( 5x - 25 \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(5x - 25) = \frac{1}{5} (5x - 25) + 5 = x - 5 + 5 = x \][/tex]
[tex]\( g(f(x)) = x \)[/tex], this part holds true.
4. Calculate [tex]\( f(g(x)) \)[/tex]:
[tex]\[ f(g(x)) = f\left( \frac{1}{5} x + 5 \right) \][/tex]
Substitute [tex]\( \frac{1}{5} x + 5 \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f\left(\frac{1}{5} x + 5\right) = 5\left(\frac{1}{5} x + 5\right) - 25 = x + 25 - 25 = x \][/tex]
[tex]\( f(g(x)) = x \)[/tex], this part also holds true.
5. List Possible Expressions:
Evaluate which of the given expressions simplifies to [tex]\( x \)[/tex]:
- [tex]\(\frac{1}{5}\left(\frac{1}{5} x + 5\right) + 5\)[/tex]
[tex]\[ \frac{1}{5}\left(\frac{1}{5} x + 5\right) + 5 = \frac{1}{25} x + 1 + 5 = \frac{1}{25} x + 6 \][/tex]
This does not simplify to [tex]\( x \)[/tex].
- [tex]\(\frac{1}{5}(5x - 25) + 5\)[/tex]
[tex]\[ \frac{1}{5}(5x - 25) + 5 = x - 5 + 5 = x \][/tex]
This simplifies to [tex]\( x \)[/tex].
- [tex]\(\frac{1}{\left(\frac{1}{5} x + 5\right)}\)[/tex]
[tex]\[ \frac{1}{\left(\frac{1}{5} x + 5\right)} = \text{This simplifies to} \frac{1}{5}x + 5 \text{ if this is the inverse} \][/tex]
This does not simplify to [tex]\( x \)[/tex].
- [tex]\( 5\left(\frac{1}{5} x + 5\right) + 5 \)[/tex]
[tex]\[ 5\left(\frac{1}{5} x + 5\right) + 5 = x + 25 + 5 = x + 30 \][/tex]
This does not simplify to [tex]\( x \)[/tex].
Based on this analysis, the second expression, [tex]\(\frac{1}{5}(5x - 25) + 5 \)[/tex], is the one that can be used to verify if [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex].
Therefore, the correct option is:
[tex]\[ \boxed{\frac{1}{5}(5x - 25) + 5} \][/tex]
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.
