Answer :
To find the inverse of the function \( y = x^2 + 4x + 4 \), we will follow a structured approach.
### Step-by-Step Solution
1. Express the function in a more manageable form:
The given function is \( y = x^2 + 4x + 4 \). We notice that this can be rewritten by completing the square:
[tex]\[ y = (x + 2)^2 \][/tex]
2. Swap \( x \) and \( y \):
To find the inverse function, we need to express \( x \) in terms of \( y \). We start by swapping \( x \) and \( y \):
[tex]\[ x = (y + 2)^2 \][/tex]
3. Solve for \( y \):
Next, we need to isolate \( y \). We do this by taking the square root of both sides:
[tex]\[ \sqrt{x} = y + 2 \][/tex]
Note that we should consider both the positive and negative square roots, but since we usually assume the principal (positive) square root unless otherwise specified, we will proceed with:
[tex]\[ \sqrt{x} = y + 2 \][/tex]
4. Isolate \( y \):
Finally, we solve for \( y \) by subtracting 2 from both sides:
[tex]\[ y = \sqrt{x} - 2 \][/tex]
### Conclusion
The inverse of the function \( y = x^2 + 4x + 4 \) is:
[tex]\[ y = \sqrt{x} - 2 \][/tex]
This is the step-by-step process to find the inverse of the given function.
### Step-by-Step Solution
1. Express the function in a more manageable form:
The given function is \( y = x^2 + 4x + 4 \). We notice that this can be rewritten by completing the square:
[tex]\[ y = (x + 2)^2 \][/tex]
2. Swap \( x \) and \( y \):
To find the inverse function, we need to express \( x \) in terms of \( y \). We start by swapping \( x \) and \( y \):
[tex]\[ x = (y + 2)^2 \][/tex]
3. Solve for \( y \):
Next, we need to isolate \( y \). We do this by taking the square root of both sides:
[tex]\[ \sqrt{x} = y + 2 \][/tex]
Note that we should consider both the positive and negative square roots, but since we usually assume the principal (positive) square root unless otherwise specified, we will proceed with:
[tex]\[ \sqrt{x} = y + 2 \][/tex]
4. Isolate \( y \):
Finally, we solve for \( y \) by subtracting 2 from both sides:
[tex]\[ y = \sqrt{x} - 2 \][/tex]
### Conclusion
The inverse of the function \( y = x^2 + 4x + 4 \) is:
[tex]\[ y = \sqrt{x} - 2 \][/tex]
This is the step-by-step process to find the inverse of the given function.
