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Sagot :
To determine the inverse of the function [tex]\( f(x) = \frac{2x - 1}{x + 3} \)[/tex] and to find the range of [tex]\( f(x) \)[/tex], we can follow these steps:
### Step 1: Find the inverse [tex]\( f^{-1}(x) \)[/tex]
1. Start with [tex]\( y = f(x) \)[/tex]:
[tex]\[ y = \frac{2x - 1}{x + 3} \][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
First, multiply both sides by [tex]\( x + 3 \)[/tex] to eliminate the fraction:
[tex]\[ y(x + 3) = 2x - 1 \][/tex]
3. Expand and rearrange the equation to isolate [tex]\( x \)[/tex]:
[tex]\[ yx + 3y = 2x - 1 \][/tex]
[tex]\[ yx - 2x = -1 - 3y \][/tex]
[tex]\[ x(y - 2) = -(3y + 1) \][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{-(3y + 1)}{y - 2} \][/tex]
This is the expression for the inverse function:
[tex]\[ f^{-1}(x) = \frac{-(3x + 1)}{x - 2} \][/tex]
### Step 2: Match the inverse with the given options
Given options for the inverse function are:
1. [tex]\( f^{-1} = \frac{1 + 3x}{2 - x} \)[/tex]
2. [tex]\( f^{-1} = \frac{2 - x}{1 + 3x} \)[/tex]
3. [tex]\( f^{-1} = \frac{1 + 2x}{3 - x} \)[/tex]
4. [tex]\( f^{-1} = \frac{1 - 3x}{2 + x} \)[/tex]
Now let's analyze and see if any of the options match our derived inverse:
[tex]\[ f^{-1}(x) = \frac{-(3x + 1)}{x - 2} \][/tex]
Unfortunately, none of the provided inverse functions match the computed inverse.
### Step 3: Determine the range of [tex]\( f(x) \)[/tex]
Since none of the options match the computed inverse, the range corresponding to each of the provided potential inverses is irrelevant here.
However, generally to find the range of the function [tex]\( f(x) \)[/tex], we can observe the behavior of the function as [tex]\( x \)[/tex] approaches its critical values (asymptotes and zeros):
[tex]\[ f(x) = \frac{2x - 1}{x + 3} \][/tex]
The function [tex]\( f(x) \)[/tex] has a vertical asymptote at [tex]\( x = -3 \)[/tex] (where the denominator is zero), causing the function to take all real values except for the value that causes a break in the function.
Thus, based on the provided information:
Inverse Function: None of the provided options match the computed inverse.
Range of [tex]\( f(x) \)[/tex]: Undefined due to the provided information mismatch.
So the final conclusion is:
- Inverse Function: None of the provided options match the computed inverse.
- Range of [tex]\( f(x) \)[/tex]: Undefined.
Let's analyze or recheck further to correct our work manually to figure if any range tied directly to any mismatch of the logic provided but the real computed worked inversely none perfectly matched thus, stating Undefined more confidently.
### Step 1: Find the inverse [tex]\( f^{-1}(x) \)[/tex]
1. Start with [tex]\( y = f(x) \)[/tex]:
[tex]\[ y = \frac{2x - 1}{x + 3} \][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
First, multiply both sides by [tex]\( x + 3 \)[/tex] to eliminate the fraction:
[tex]\[ y(x + 3) = 2x - 1 \][/tex]
3. Expand and rearrange the equation to isolate [tex]\( x \)[/tex]:
[tex]\[ yx + 3y = 2x - 1 \][/tex]
[tex]\[ yx - 2x = -1 - 3y \][/tex]
[tex]\[ x(y - 2) = -(3y + 1) \][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{-(3y + 1)}{y - 2} \][/tex]
This is the expression for the inverse function:
[tex]\[ f^{-1}(x) = \frac{-(3x + 1)}{x - 2} \][/tex]
### Step 2: Match the inverse with the given options
Given options for the inverse function are:
1. [tex]\( f^{-1} = \frac{1 + 3x}{2 - x} \)[/tex]
2. [tex]\( f^{-1} = \frac{2 - x}{1 + 3x} \)[/tex]
3. [tex]\( f^{-1} = \frac{1 + 2x}{3 - x} \)[/tex]
4. [tex]\( f^{-1} = \frac{1 - 3x}{2 + x} \)[/tex]
Now let's analyze and see if any of the options match our derived inverse:
[tex]\[ f^{-1}(x) = \frac{-(3x + 1)}{x - 2} \][/tex]
Unfortunately, none of the provided inverse functions match the computed inverse.
### Step 3: Determine the range of [tex]\( f(x) \)[/tex]
Since none of the options match the computed inverse, the range corresponding to each of the provided potential inverses is irrelevant here.
However, generally to find the range of the function [tex]\( f(x) \)[/tex], we can observe the behavior of the function as [tex]\( x \)[/tex] approaches its critical values (asymptotes and zeros):
[tex]\[ f(x) = \frac{2x - 1}{x + 3} \][/tex]
The function [tex]\( f(x) \)[/tex] has a vertical asymptote at [tex]\( x = -3 \)[/tex] (where the denominator is zero), causing the function to take all real values except for the value that causes a break in the function.
Thus, based on the provided information:
Inverse Function: None of the provided options match the computed inverse.
Range of [tex]\( f(x) \)[/tex]: Undefined due to the provided information mismatch.
So the final conclusion is:
- Inverse Function: None of the provided options match the computed inverse.
- Range of [tex]\( f(x) \)[/tex]: Undefined.
Let's analyze or recheck further to correct our work manually to figure if any range tied directly to any mismatch of the logic provided but the real computed worked inversely none perfectly matched thus, stating Undefined more confidently.
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