Finding the Inverse of a Function
To find the inverse of the function \( f(x) = 2(x + 4)^2 - 5 \) for \( x \leq -4 \), we'll follow these steps:
1. Replace \( f(x) \) with \( y \):
\[ y = 2(x + 4)^2 - 5 \]
2. Swap \( x \) and \( y \), because the inverse function \( f^{-1}(x) \) will take the output of \( f \) (which we're calling \( y \)) and produce the original input \( x \):
\[ x = 2(y + 4)^2 - 5 \]
3. Solve this equation for \( y \) to find \( f^{-1}(x) \):
\[ x + 5 = 2(y + 4)^2 \]
\[ \frac{x + 5}{2} = (y + 4)^2 \]
Now we take the square root of both sides. Since we know that \( x \leq -4 \), that means the function is dealing with the left half of the parabola, where \( y \) must be less than or equal to -4 since it's decreasing at that part. So, we choose the negative square root to maintain the function inverse:
\[ \sqrt{\frac{x + 5}{2}} = y + 4 \]
\[ y = -4 \pm \sqrt{\frac{x + 5}{2}} \]
Since \( y \leq -4 \), we use the negative square root:
\[ y = -4 - \sqrt{\frac{x + 5}{2}} \]
So the inverse function \( f^{-1}(x) \) is:
\[ f^{-1}(x) = -4 - \sqrt{\frac{x + 5}{2}} \]
To check the solution algebraically, you would compose \( f \) and \( f^{-1} \) and ensure that the composition equals \( x \) and vice versa. Graphically, the function \( f \) and its inverse \( f^{-1} \) should be reflections of each other across the line \( y = x \). Unfortunately, as an AI, I can't confirm the graphic representation directly, but you can check this by plotting both functions on graph paper or a graphing utility.
