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Sagot :
To determine which function is the inverse of [tex]\( f(x) = \frac{1}{2}x + 5 \)[/tex], we will follow the steps for finding the inverse of a function and then verify it.
### Step-by-Step Solution
1. Find the Inverse Algebraically:
- Start with the function [tex]\( f(x) = \frac{1}{2}x + 5 \)[/tex].
- Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = \frac{1}{2}x + 5 \][/tex]
- Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse function:
[tex]\[ x = \frac{1}{2}y + 5 \][/tex]
- Solve this equation for [tex]\( y \)[/tex]:
- Subtract 5 from both sides:
[tex]\[ x - 5 = \frac{1}{2}y \][/tex]
- Multiply both sides by 2 to solve for [tex]\( y \)[/tex]:
[tex]\[ 2(x - 5) = y \][/tex]
- Simplify:
[tex]\[ y = 2x - 10 \][/tex]
- Thus, the inverse function is [tex]\( f^{-1}(x) = 2x - 10 \)[/tex].
2. Verify the Correct Inverse Function:
- We need to confirm that [tex]\( f(f^{-1}(x)) = x \)[/tex] and [tex]\( f^{-1}(f(x)) = x \)[/tex].
- Let's check with [tex]\( f^{-1}(x) = 2x - 10 \)[/tex]:
1. Evaluate [tex]\( f(f^{-1}(x)) \)[/tex]:
[tex]\[ f(2x - 10) = \frac{1}{2}(2x - 10) + 5 \][/tex]
Simplify:
[tex]\[ f(2x - 10) = x - 5 + 5 = x \][/tex]
So, [tex]\( f(f^{-1}(x)) = x \)[/tex].
2. Evaluate [tex]\( f^{-1}(f(x)) \)[/tex]:
[tex]\[ f^{-1}\left( \frac{1}{2}x + 5 \right) = 2 \left( \frac{1}{2}x + 5 \right) - 10 \][/tex]
Simplify:
[tex]\[ f^{-1}\left( \frac{1}{2}x + 5 \right) = x + 10 - 10 = x \][/tex]
So, [tex]\( f^{-1}(f(x)) = x \)[/tex].
Since both conditions hold true, the correct inverse function is [tex]\( f^{-1}(x) = 2x - 10 \)[/tex].
### Conclusion
The function that is the inverse of [tex]\( f(x) = \frac{1}{2}x + 5 \)[/tex] is:
[tex]\[ f^{-1}(x) = 2x - 10 \][/tex]
Thus, the correct choice is:
[tex]\[ f^{-1}(x) = 2x - 10 \][/tex]
### Step-by-Step Solution
1. Find the Inverse Algebraically:
- Start with the function [tex]\( f(x) = \frac{1}{2}x + 5 \)[/tex].
- Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = \frac{1}{2}x + 5 \][/tex]
- Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse function:
[tex]\[ x = \frac{1}{2}y + 5 \][/tex]
- Solve this equation for [tex]\( y \)[/tex]:
- Subtract 5 from both sides:
[tex]\[ x - 5 = \frac{1}{2}y \][/tex]
- Multiply both sides by 2 to solve for [tex]\( y \)[/tex]:
[tex]\[ 2(x - 5) = y \][/tex]
- Simplify:
[tex]\[ y = 2x - 10 \][/tex]
- Thus, the inverse function is [tex]\( f^{-1}(x) = 2x - 10 \)[/tex].
2. Verify the Correct Inverse Function:
- We need to confirm that [tex]\( f(f^{-1}(x)) = x \)[/tex] and [tex]\( f^{-1}(f(x)) = x \)[/tex].
- Let's check with [tex]\( f^{-1}(x) = 2x - 10 \)[/tex]:
1. Evaluate [tex]\( f(f^{-1}(x)) \)[/tex]:
[tex]\[ f(2x - 10) = \frac{1}{2}(2x - 10) + 5 \][/tex]
Simplify:
[tex]\[ f(2x - 10) = x - 5 + 5 = x \][/tex]
So, [tex]\( f(f^{-1}(x)) = x \)[/tex].
2. Evaluate [tex]\( f^{-1}(f(x)) \)[/tex]:
[tex]\[ f^{-1}\left( \frac{1}{2}x + 5 \right) = 2 \left( \frac{1}{2}x + 5 \right) - 10 \][/tex]
Simplify:
[tex]\[ f^{-1}\left( \frac{1}{2}x + 5 \right) = x + 10 - 10 = x \][/tex]
So, [tex]\( f^{-1}(f(x)) = x \)[/tex].
Since both conditions hold true, the correct inverse function is [tex]\( f^{-1}(x) = 2x - 10 \)[/tex].
### Conclusion
The function that is the inverse of [tex]\( f(x) = \frac{1}{2}x + 5 \)[/tex] is:
[tex]\[ f^{-1}(x) = 2x - 10 \][/tex]
Thus, the correct choice is:
[tex]\[ f^{-1}(x) = 2x - 10 \][/tex]
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