Discover the answers to your questions at Westonci.ca, where experts share their knowledge and insights with you. Explore in-depth answers to your questions from a knowledgeable community of experts across different fields. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
To determine which pair of expressions represents inverse functions, we need to verify that for a given pair of functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], the following conditions are met:
1. [tex]\( f(g(x)) = x \)[/tex]
2. [tex]\( g(f(x)) = x \)[/tex]
Let's analyze each option step-by-step:
Option A:
[tex]\[ f(x) = \frac{4 - 3x}{4x - 2}, \quad g(x) = \frac{x + 2}{x - 2} \][/tex]
1. [tex]\( f(g(x)) = f\left(\frac{x + 2}{x - 2}\right) \)[/tex]:
Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f\left(\frac{x + 2}{x - 2}\right) = \frac{4 - 3\left(\frac{x + 2}{x - 2}\right)}{4\left(\frac{x + 2}{x - 2}\right) - 2} \][/tex]
2. [tex]\( g(f(x)) = g\left(\frac{4 - 3x}{4x - 2}\right) \)[/tex]:
Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g\left(\frac{4 - 3x}{4x - 2}\right) = \frac{\left(\frac{4 - 3x}{4x - 2}\right) + 2}{\left(\frac{4 - 3x}{4x - 2}\right) - 2} \][/tex]
After simplifying, we find that [tex]\( f(g(x)) \neq x \)[/tex] and [tex]\( g(f(x)) \neq x \)[/tex]. Therefore, these functions are not inverses.
Option B:
[tex]\[ f(x) = \frac{x + 3}{4x - 2}, \quad g(x) = \frac{2x + 3}{4x - 1} \][/tex]
1. [tex]\( f(g(x)) = f\left(\frac{2x + 3}{4x - 1}\right) \)[/tex]:
Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f\left(\frac{2x + 3}{4x - 1}\right) = \frac{\frac{2x + 3}{4x - 1} + 3}{4\left(\frac{2x + 3}{4x - 1}\right) - 2} = x \][/tex]
2. [tex]\( g(f(x)) = g\left(\frac{x + 3}{4x - 2}\right) \)[/tex]:
Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g\left(\frac{x + 3}{4x - 2}\right) = \frac{2\left(\frac{x + 3}{4x - 2}\right) + 3}{4\left(\frac{x + 3}{4x - 2}\right) - 1} = x \][/tex]
These simplifications confirm that [tex]\( f(g(x)) = x \)[/tex] and [tex]\( g(f(x)) = x \)[/tex]. Therefore, [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are indeed inverse functions.
Option C:
[tex]\[ f(x) = \frac{4x + 2}{x - 3}, \quad g(x) = \frac{5x + 3}{4x - 2} \][/tex]
1. [tex]\( f(g(x)) = f\left(\frac{5x + 3}{4x - 2}\right) \)[/tex]:
Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f\left(\frac{5x + 3}{4x - 2}\right) = \frac{4\left(\frac{5x + 3}{4x - 2}\right) + 2}{\left(\frac{5x + 3}{4x - 2}\right) - 3} \][/tex]
2. [tex]\( g(f(x)) = g\left(\frac{4x + 2}{x - 3}\right) \)[/tex]:
Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g\left(\frac{4x + 2}{x - 3}\right) = \frac{5\left(\frac{4x + 2}{x - 3}\right) + 3}{4\left(\frac{4x + 2}{x - 3}\right) - 2} \][/tex]
After simplification, it is concluded that these expressions do not satisfy the inverse conditions.
Option D:
[tex]\[ f(x) = 2x + 5, \quad g(x) = 2 + 5x \][/tex]
1. [tex]\( f(g(x)) = f(2 + 5x) \)[/tex]:
Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(2 + 5x) = 2(2 + 5x) + 5 = 4 + 10x + 5 = 9 + 10x \neq x \][/tex]
2. [tex]\( g(f(x)) = g(2x + 5) \)[/tex]:
Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(2x + 5) = 2 + 5(2x + 5) = 2 + 10x + 25 = 27 + 10x \neq x \][/tex]
Therefore, these functions are also not inverses.
After reviewing all options:
The pair of expressions that represents inverse functions are those in Option B:
[tex]\[ \boxed{2} \][/tex]
1. [tex]\( f(g(x)) = x \)[/tex]
2. [tex]\( g(f(x)) = x \)[/tex]
Let's analyze each option step-by-step:
Option A:
[tex]\[ f(x) = \frac{4 - 3x}{4x - 2}, \quad g(x) = \frac{x + 2}{x - 2} \][/tex]
1. [tex]\( f(g(x)) = f\left(\frac{x + 2}{x - 2}\right) \)[/tex]:
Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f\left(\frac{x + 2}{x - 2}\right) = \frac{4 - 3\left(\frac{x + 2}{x - 2}\right)}{4\left(\frac{x + 2}{x - 2}\right) - 2} \][/tex]
2. [tex]\( g(f(x)) = g\left(\frac{4 - 3x}{4x - 2}\right) \)[/tex]:
Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g\left(\frac{4 - 3x}{4x - 2}\right) = \frac{\left(\frac{4 - 3x}{4x - 2}\right) + 2}{\left(\frac{4 - 3x}{4x - 2}\right) - 2} \][/tex]
After simplifying, we find that [tex]\( f(g(x)) \neq x \)[/tex] and [tex]\( g(f(x)) \neq x \)[/tex]. Therefore, these functions are not inverses.
Option B:
[tex]\[ f(x) = \frac{x + 3}{4x - 2}, \quad g(x) = \frac{2x + 3}{4x - 1} \][/tex]
1. [tex]\( f(g(x)) = f\left(\frac{2x + 3}{4x - 1}\right) \)[/tex]:
Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f\left(\frac{2x + 3}{4x - 1}\right) = \frac{\frac{2x + 3}{4x - 1} + 3}{4\left(\frac{2x + 3}{4x - 1}\right) - 2} = x \][/tex]
2. [tex]\( g(f(x)) = g\left(\frac{x + 3}{4x - 2}\right) \)[/tex]:
Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g\left(\frac{x + 3}{4x - 2}\right) = \frac{2\left(\frac{x + 3}{4x - 2}\right) + 3}{4\left(\frac{x + 3}{4x - 2}\right) - 1} = x \][/tex]
These simplifications confirm that [tex]\( f(g(x)) = x \)[/tex] and [tex]\( g(f(x)) = x \)[/tex]. Therefore, [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are indeed inverse functions.
Option C:
[tex]\[ f(x) = \frac{4x + 2}{x - 3}, \quad g(x) = \frac{5x + 3}{4x - 2} \][/tex]
1. [tex]\( f(g(x)) = f\left(\frac{5x + 3}{4x - 2}\right) \)[/tex]:
Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f\left(\frac{5x + 3}{4x - 2}\right) = \frac{4\left(\frac{5x + 3}{4x - 2}\right) + 2}{\left(\frac{5x + 3}{4x - 2}\right) - 3} \][/tex]
2. [tex]\( g(f(x)) = g\left(\frac{4x + 2}{x - 3}\right) \)[/tex]:
Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g\left(\frac{4x + 2}{x - 3}\right) = \frac{5\left(\frac{4x + 2}{x - 3}\right) + 3}{4\left(\frac{4x + 2}{x - 3}\right) - 2} \][/tex]
After simplification, it is concluded that these expressions do not satisfy the inverse conditions.
Option D:
[tex]\[ f(x) = 2x + 5, \quad g(x) = 2 + 5x \][/tex]
1. [tex]\( f(g(x)) = f(2 + 5x) \)[/tex]:
Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(2 + 5x) = 2(2 + 5x) + 5 = 4 + 10x + 5 = 9 + 10x \neq x \][/tex]
2. [tex]\( g(f(x)) = g(2x + 5) \)[/tex]:
Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(2x + 5) = 2 + 5(2x + 5) = 2 + 10x + 25 = 27 + 10x \neq x \][/tex]
Therefore, these functions are also not inverses.
After reviewing all options:
The pair of expressions that represents inverse functions are those in Option B:
[tex]\[ \boxed{2} \][/tex]
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.
