Consider the function𝑓(𝑥)=𝑥2−6,𝑥≤0Find the inverse function of 𝑓.

Sure, let's find the inverse function of $f(x) = x^2 - 6$ for $x \leq 0$ step by step.

Step 1: Write the function in terms of $y$

First, we write the function $f(x) = x^2 - 6$ in terms of $y$: $y = x^2 - 6$

Step 2: Solve for $x$ in terms of $y$

Next, we solve this equation for $x$: $y + 6 = x^2$ $x = \pm \sqrt{y + 6}$

Step 3: Consider the domain restriction

Since the original function $f(x)$ is defined for $x \leq 0$, we need to consider the negative square root to maintain the domain restriction: $x = -\sqrt{y + 6}$

Step 4: Express the inverse function

Now, we express the inverse function $f^{-1}(y)$ by swapping $x$ and $y$: $f^{-1}(y) = -\sqrt{y + 6}$

Step 5: Verify the inverse function

To verify, we can check if $f(f^{-1}(y)) = y$ and $f^{-1}(f(x)) = x$:

1. $f(f^{-1}(y))$: $f(f^{-1}(y)) = f(-\sqrt{y + 6})$ $= (-\sqrt{y + 6})^2 - 6$ $= (y + 6) - 6$ $= y$

2. $f^{-1}(f(x))$: $f^{-1}(f(x)) = f^{-1}(x^2 - 6)$ $= -\sqrt{(x^2 - 6) + 6}$ $= -\sqrt{x^2}$ $= -|x|$ Since $x \leq 0$, $-|x| = x$.

Thus, the inverse function is correctly found.

Final Answer

The inverse function of $f(x) = x^2 - 6$ for $x \leq 0$ is: $f^{-1}(y) = -\sqrt{y + 6}$