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Sagot :
To determine which pair of expressions represents inverse functions, we'll first recall the property of inverse functions. If [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are inverse functions, then [tex]\( f(g(x)) = x \)[/tex] and [tex]\( g(f(x)) = x \)[/tex] must hold true. Let's analyze the given options step-by-step.
### Option A: [tex]\(\frac{4-3x}{4x-2}\)[/tex] and [tex]\(\frac{x+2}{x-2}\)[/tex]
Let's denote:
[tex]\( f(x) = \frac{4-3x}{4x-2} \)[/tex]
[tex]\( g(x) = \frac{x+2}{x-2} \)[/tex]
For these to be inverse functions:
1. [tex]\( f(g(x)) = x \)[/tex]
2. [tex]\( g(f(x)) = x \)[/tex]
1. Calculate [tex]\( f(g(x)) \)[/tex]:
[tex]\[ f(g(x)) = f\left( \frac{x+2}{x-2} \right) = \frac{4 - 3 \cdot \frac{x+2}{x-2}}{4 \cdot \frac{x+2}{x-2} - 2} \][/tex]
2. Calculate [tex]\( g(f(x)) \)[/tex]:
[tex]\[ g(f(x)) = g\left( \frac{4-3x}{4x-2} \right) = \frac{\frac{4-3x}{4x-2} + 2}{\frac{4-3x}{4x-2} - 2} \][/tex]
By simplifying, we need to check if both [tex]\( f(g(x)) = x \)[/tex] and [tex]\( g(f(x)) = x \)[/tex]. However, when these simplifications are carried out, neither of them equals [tex]\( x \)[/tex], thus they are not inverses.
### Option B: [tex]\(\frac{x+3}{4x-2}\)[/tex] and [tex]\(\frac{2z+3}{4x-1}\)[/tex]
Denote:
[tex]\( f(x) = \frac{x+3}{4x-2} \)[/tex]
[tex]\( g(x) = \frac{2z+3}{4x-1} \)[/tex]
Here, the expression for [tex]\( g(x) \)[/tex] has different variables ([tex]\( z \)[/tex] and [tex]\( x \)[/tex]) which makes it impractical to verify inverse relationships. Additionally, different denominators in [tex]\( g(x) \)[/tex] suggest they are not formatted to be related inverses.
### Option C: [tex]\(\frac{4x+2}{x-3}\)[/tex] and [tex]\(\frac{5z+3}{4x-2}\)[/tex]
Denote:
[tex]\( f(x) = \frac{4x+2}{x-3} \)[/tex]
[tex]\( g(x) = \frac{5z+3}{4x-2} \)[/tex]
Similarly to option B, mismatched variables ([tex]\( z \)[/tex] and [tex]\( x \)[/tex]) and different constructions of the fractional terms indicate that performing verification for inverse properties isn’t feasible or structured correctly.
### Option D: [tex]\(2x+5\)[/tex] and [tex]\(2+5x\)[/tex]
Denote:
[tex]\( f(x) = 2x+5 \)[/tex]
[tex]\( g(x) = 2+5x \)[/tex]
For [tex]\( g(x) \)[/tex] to be the inverse of [tex]\( f(x) \)[/tex], the composition should yield the identity function:
1. [tex]\( f(g(x)) = 2(2+5x) + 5 \)[/tex]
2. [tex]\( g(f(x)) = 2 + 5(2x+5) \)[/tex]
Calculations:
1. [tex]\( f(g(x)) = 2(2 + 5x) + 5 = 4 + 10x + 5 = 9 + 10x \)[/tex]
2. [tex]\( g(f(x)) = 2 + 5(2x + 5) = 2 + 10x + 25 = 27 + 10x \)[/tex]
Neither composition returns [tex]\( x \)[/tex] as expected from the inverse function definition, hence these are not inverse functions.
Based on the analysis above, no pair among the given choices represents inverse functions.
Thus, the correct answer is:
[tex]\[ \boxed{\text{None of the given pairs are inverse functions.}} \][/tex]
### Option A: [tex]\(\frac{4-3x}{4x-2}\)[/tex] and [tex]\(\frac{x+2}{x-2}\)[/tex]
Let's denote:
[tex]\( f(x) = \frac{4-3x}{4x-2} \)[/tex]
[tex]\( g(x) = \frac{x+2}{x-2} \)[/tex]
For these to be inverse functions:
1. [tex]\( f(g(x)) = x \)[/tex]
2. [tex]\( g(f(x)) = x \)[/tex]
1. Calculate [tex]\( f(g(x)) \)[/tex]:
[tex]\[ f(g(x)) = f\left( \frac{x+2}{x-2} \right) = \frac{4 - 3 \cdot \frac{x+2}{x-2}}{4 \cdot \frac{x+2}{x-2} - 2} \][/tex]
2. Calculate [tex]\( g(f(x)) \)[/tex]:
[tex]\[ g(f(x)) = g\left( \frac{4-3x}{4x-2} \right) = \frac{\frac{4-3x}{4x-2} + 2}{\frac{4-3x}{4x-2} - 2} \][/tex]
By simplifying, we need to check if both [tex]\( f(g(x)) = x \)[/tex] and [tex]\( g(f(x)) = x \)[/tex]. However, when these simplifications are carried out, neither of them equals [tex]\( x \)[/tex], thus they are not inverses.
### Option B: [tex]\(\frac{x+3}{4x-2}\)[/tex] and [tex]\(\frac{2z+3}{4x-1}\)[/tex]
Denote:
[tex]\( f(x) = \frac{x+3}{4x-2} \)[/tex]
[tex]\( g(x) = \frac{2z+3}{4x-1} \)[/tex]
Here, the expression for [tex]\( g(x) \)[/tex] has different variables ([tex]\( z \)[/tex] and [tex]\( x \)[/tex]) which makes it impractical to verify inverse relationships. Additionally, different denominators in [tex]\( g(x) \)[/tex] suggest they are not formatted to be related inverses.
### Option C: [tex]\(\frac{4x+2}{x-3}\)[/tex] and [tex]\(\frac{5z+3}{4x-2}\)[/tex]
Denote:
[tex]\( f(x) = \frac{4x+2}{x-3} \)[/tex]
[tex]\( g(x) = \frac{5z+3}{4x-2} \)[/tex]
Similarly to option B, mismatched variables ([tex]\( z \)[/tex] and [tex]\( x \)[/tex]) and different constructions of the fractional terms indicate that performing verification for inverse properties isn’t feasible or structured correctly.
### Option D: [tex]\(2x+5\)[/tex] and [tex]\(2+5x\)[/tex]
Denote:
[tex]\( f(x) = 2x+5 \)[/tex]
[tex]\( g(x) = 2+5x \)[/tex]
For [tex]\( g(x) \)[/tex] to be the inverse of [tex]\( f(x) \)[/tex], the composition should yield the identity function:
1. [tex]\( f(g(x)) = 2(2+5x) + 5 \)[/tex]
2. [tex]\( g(f(x)) = 2 + 5(2x+5) \)[/tex]
Calculations:
1. [tex]\( f(g(x)) = 2(2 + 5x) + 5 = 4 + 10x + 5 = 9 + 10x \)[/tex]
2. [tex]\( g(f(x)) = 2 + 5(2x + 5) = 2 + 10x + 25 = 27 + 10x \)[/tex]
Neither composition returns [tex]\( x \)[/tex] as expected from the inverse function definition, hence these are not inverse functions.
Based on the analysis above, no pair among the given choices represents inverse functions.
Thus, the correct answer is:
[tex]\[ \boxed{\text{None of the given pairs are inverse functions.}} \][/tex]
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