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Sagot :
To determine the correctness of a student's work in comparing a function and its inverse, let's break down the steps for graphing the inverse of an absolute value function \( f(x) = |x| \) after restricting its domain, and verifying the result.
### Step 1: Understanding the Absolute Value Function
The absolute value function \( f(x) = |x| \) consists of two pieces:
- For \( x \geq 0 \), \( f(x) = x \)
- For \( x < 0 \), \( f(x) = -x \)
Graphically, this represents a V-shaped graph that opens upwards with its vertex at the origin (0,0).
### Step 2: Restricting the Domain
To create an inverse function, we need a one-to-one function (each output is mapped to by only one input). Since \( f(x) = |x| \) is not one-to-one over its entire domain (\( -\infty < x < \infty \)), we must restrict its domain.
Typically, we restrict the domain to \( x \geq 0 \) to make it one-to-one. Thus, we consider:
[tex]\[ f(x) = x \text{ for } x \geq 0 \][/tex]
### Step 3: Finding the Inverse Function
For the restricted function \( f(x) = x \), we find the inverse \( f^{-1}(x) \):
1. Start with \( y = f(x) = x \).
2. Swap \( x \) and \( y \): \( x = y \).
3. Solve for \( y \): \( y = x \).
Therefore, \( f^{-1}(x) = x \).
### Step 4: Verifying the Inverse Relationship
To verify that two functions \( f(x) \) and \( f^{-1}(x) \) are indeed inverses of each other, their graphs should be symmetric about the line \( y = x \). Thus:
- Graph \( f(x) = x \) for \( x \geq 0 \).
- Graph \( f^{-1}(x) = x \).
Both graphs will coincide with the line \( y = x \) for \( x \geq 0 \).
### Step 5: Dashed Line Verification
To visually confirm that \( f \) and \( f^{-1} \) are inverses, include a dashed line \( y = x \) (45-degree line). If both functions are symmetric about this line, then your functions are correctly graphed as inverses.
### Who is Correct?
Given these steps, the correct student should have:
- Graphed the function \( f(x) = x \) (starting from the origin and extending to the right).
- Graphed the inverse function \( f^{-1}(x) = x \) correctly, which will overlap with the original function on the restricted domain.
- Included a dashed line \( y = x \) confirming the symmetry about this line.
Verify these visual elements in each student's graph to determine who has correctly completed the task. The student's graph that adheres to these steps will be the correct one.
### Step 1: Understanding the Absolute Value Function
The absolute value function \( f(x) = |x| \) consists of two pieces:
- For \( x \geq 0 \), \( f(x) = x \)
- For \( x < 0 \), \( f(x) = -x \)
Graphically, this represents a V-shaped graph that opens upwards with its vertex at the origin (0,0).
### Step 2: Restricting the Domain
To create an inverse function, we need a one-to-one function (each output is mapped to by only one input). Since \( f(x) = |x| \) is not one-to-one over its entire domain (\( -\infty < x < \infty \)), we must restrict its domain.
Typically, we restrict the domain to \( x \geq 0 \) to make it one-to-one. Thus, we consider:
[tex]\[ f(x) = x \text{ for } x \geq 0 \][/tex]
### Step 3: Finding the Inverse Function
For the restricted function \( f(x) = x \), we find the inverse \( f^{-1}(x) \):
1. Start with \( y = f(x) = x \).
2. Swap \( x \) and \( y \): \( x = y \).
3. Solve for \( y \): \( y = x \).
Therefore, \( f^{-1}(x) = x \).
### Step 4: Verifying the Inverse Relationship
To verify that two functions \( f(x) \) and \( f^{-1}(x) \) are indeed inverses of each other, their graphs should be symmetric about the line \( y = x \). Thus:
- Graph \( f(x) = x \) for \( x \geq 0 \).
- Graph \( f^{-1}(x) = x \).
Both graphs will coincide with the line \( y = x \) for \( x \geq 0 \).
### Step 5: Dashed Line Verification
To visually confirm that \( f \) and \( f^{-1} \) are inverses, include a dashed line \( y = x \) (45-degree line). If both functions are symmetric about this line, then your functions are correctly graphed as inverses.
### Who is Correct?
Given these steps, the correct student should have:
- Graphed the function \( f(x) = x \) (starting from the origin and extending to the right).
- Graphed the inverse function \( f^{-1}(x) = x \) correctly, which will overlap with the original function on the restricted domain.
- Included a dashed line \( y = x \) confirming the symmetry about this line.
Verify these visual elements in each student's graph to determine who has correctly completed the task. The student's graph that adheres to these steps will be the correct one.
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