Question - Finding the Inverse of a Function
Solution:
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.
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