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Part 1
Find the inverse of the function. Is the inverse a function?
f(x)equalsleft parenthesis x plus 4 right parenthesis squared
To find the inverse of the function \( f(x) = (x+4)^2 \), we can follow these steps:
1. Substitute \( f(x) = y \)
2. Swap x and y: \( x = (y+4)^2 \)
3. Solve for y:
\[ x = (y+4)^2 \]
\[ \sqrt{x} = y+4 \]
\[ y = \sqrt{x} - 4 \]
Therefore, the inverse function of \( f(x) = (x+4)^2 \) is \( f^{-1}(x) = \sqrt{x} - 4 \).
To determine if the inverse is a function, we need to check if it passes the vertical line test. Since \( f^{-1}(x) = \sqrt{x} - 4 \) is a square root function, it is indeed a function because each input (x) corresponds to only one output (y). Thus, the inverse is a function.