Inverse Function: F(x) = 2(x+5)^2-2 + Graphing

Let's dive into finding the inverse of the function $f(x) = 2(x+5)^2 - 2$, where $x leq -5$, and then graph both the original function and its inverse. This is a classic problem in mathematics that combines algebraic manipulation with graphical representation. Understanding how to find inverse functions and visualize them is super useful for calculus and beyond. So, let's get started!

1. Understanding the Original Function

Before we jump into finding the inverse, let's understand the original function, $f(x) = 2(x+5)^2 - 2$, where $x leq -5$. This is a quadratic function in vertex form. The vertex form of a quadratic function is given by $f(x) = a(x-h)^2 + k$, where $(h, k)$ is the vertex of the parabola. In our case, $a = 2$, $h = -5$, and $k = -2$. This tells us that the vertex of the parabola is at $(-5, -2)$.

Because $a = 2$ is positive, the parabola opens upwards. The domain restriction $x leq -5$ tells us that we are only considering the left half of the parabola, starting from the vertex. This restriction is crucial because it ensures that the original function is one-to-one, which is a requirement for the existence of an inverse function. If we didn't have this restriction, the function would not be one-to-one over its entire domain, and thus, we wouldn't be able to find a unique inverse.

To visualize this, imagine a U-shaped parabola. Now, only consider the part of the 'U' that is to the left of the vertical line $x = -5$. This is the portion of the function we are interested in. The vertex $(-5, -2)$ is the lowest point on this section of the parabola.

Now that we understand the behavior and characteristics of the function $f(x) = 2(x+5)^2 - 2$ with the given domain restriction, we can proceed with finding its inverse. Remember, understanding the original function is key to understanding its inverse!

2. Finding the Inverse Function

Now, let's find the inverse function, $f^{-1}(x)$. Finding the inverse involves swapping $x$ and $y$ (where $y = f(x)$) and then solving for $y$. Here's how we do it step-by-step:

1. Replace $f(x)$ with $y$: We start by rewriting the function as $y = 2(x+5)^2 - 2$.

2. Swap $x$ and $y$: Next, we swap $x$ and $y$ to get $x = 2(y+5)^2 - 2$.

3. Solve for $y$: Now, we need to isolate $y$ in the equation. Let's go through the algebraic steps:

   • Add 2 to both sides: $x + 2 = 2(y+5)^2$
   • Divide both sides by 2: $\frac{x+2}{2} = (y+5)^2$
   • Take the square root of both sides: $\pm\sqrt{\frac{x+2}{2}} = y+5$
   • Subtract 5 from both sides: $y = -5 \pm \sqrt{\frac{x+2}{2}}$

4. Determine the correct sign: Since the original function is defined for $x leq -5$, the range of the inverse function will also be $y leq -5$. This means we need to choose the negative square root to ensure that the inverse function stays within the correct range. If we chose the positive square root, the values of $y$ would be greater than $-5$, which contradicts the original domain restriction.

Therefore, the inverse function is:

$f^{-1}(x) = -5 - \sqrt{\frac{x+2}{2}}$

Important Note: The domain of the inverse function is determined by the range of the original function. Since the vertex of the original parabola is at $(-5, -2)$ and it opens upwards, the range of $f(x)$ is $[-2, \infty)$. Therefore, the domain of $f^{-1}(x)$ is $[-2, \infty)$. This is because you can only take the square root of non-negative numbers.

So, to recap, finding the inverse function involved algebraic manipulation and careful consideration of the domain and range to ensure we choose the correct sign for the square root. Understanding these steps is crucial for solving similar problems in calculus and other advanced math courses.

3. Graphing the Function and Its Inverse

Now that we have both the original function and its inverse, let's discuss how to graph them. Graphing the function and its inverse can give us a visual understanding of their relationship.

1. Graphing the Original Function: $f(x) = 2(x+5)^2 - 2$ for $x leq -5$

   • Plot the vertex: The vertex is at $(-5, -2)$.
   • Choose additional points: Since we only consider $x leq -5$, let's choose $x = -6$ and $x = -7$.
     • $f(-6) = 2(-6+5)^2 - 2 = 2(-1)^2 - 2 = 0$
     • $f(-7) = 2(-7+5)^2 - 2 = 2(-2)^2 - 2 = 6$
   • Plot these points: $(-6, 0)$ and $(-7, 6)$.
   • Draw the parabola: Sketch the left half of the parabola passing through these points, starting from the vertex.

2. Graphing the Inverse Function: $f^{-1}(x) = -5 - \sqrt{\frac{x+2}{2}}$ for $x geq -2$

   • Consider the domain: The domain is $x geq -2$, so we start at $x = -2$.
   • Choose additional points: Let's choose $x = 0$ and $x = 6$.
     • $f^{-1}(0) = -5 - \sqrt{\frac{0+2}{2}} = -5 - \sqrt{1} = -6$
     • $f^{-1}(6) = -5 - \sqrt{\frac{6+2}{2}} = -5 - \sqrt{4} = -7$
   • Plot these points: $(-2, -5)$, $(0, -6)$, and $(6, -7)$.
   • Draw the curve: Sketch the curve passing through these points, starting from $(-2, -5)$.

3. The Line of Symmetry: The graphs of a function and its inverse are symmetric with respect to the line $y = x$. This means that if you were to fold the graph along the line $y = x$, the original function and its inverse would overlap. This symmetry is a visual confirmation that we have correctly found the inverse function.

When you plot these functions, you'll notice that the inverse function is a reflection of the original function across the line $y = x$. This symmetry is a key property of inverse functions and provides a visual way to check if your algebraic manipulations are correct. Make sure you have graph paper handy to plot the points accurately and see the symmetry clearly!

4. Verifying the Inverse Function

To verify the inverse function, we can use the property that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$. Let's check both:

1. Checking $f(f^{-1}(x)) = x$: We need to substitute $f^{-1}(x)$ into $f(x)$:

   $f(f^{-1}(x)) = 2((-5 - \sqrt{\frac{x+2}{2}}) + 5)^2 - 2$

   $f(f^{-1}(x)) = 2(-\sqrt{\frac{x+2}{2}})^2 - 2$

   $f(f^{-1}(x)) = 2(\frac{x+2}{2}) - 2$

   $f(f^{-1}(x)) = (x+2) - 2$

   $f(f^{-1}(x)) = x$

2. Checking $f^{-1}(f(x)) = x$: We need to substitute $f(x)$ into $f^{-1}(x)$:

   $f^{-1}(f(x)) = -5 - \sqrt{\frac{2(x+5)^2 - 2 + 2}{2}}$

   $f^{-1}(f(x)) = -5 - \sqrt{\frac{2(x+5)^2}{2}}$

   $f^{-1}(f(x)) = -5 - \sqrt{(x+5)^2}$

   Since $x leq -5$, $(x+5)$ is negative, so $\sqrt{(x+5)^2} = -(x+5)$.

   $f^{-1}(f(x)) = -5 - (-(x+5))$

   $f^{-1}(f(x)) = -5 + (x+5)$

   $f^{-1}(f(x)) = x$

Since both $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ hold true, we have verified that $f^{-1}(x)$ is indeed the inverse of $f(x)$. This verification step is crucial in confirming the accuracy of our result, especially when dealing with complex functions. Always double-check to make sure your inverse function is correct!

Conclusion

In this article, we found the inverse of the function $f(x) = 2(x+5)^2 - 2$, where $x leq -5$, and discussed how to graph both the original function and its inverse. We found that the inverse function is $f^{-1}(x) = -5 - \sqrt{\frac{x+2}{2}}$ with a domain of $x geq -2$. Additionally, we verified our result by showing that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

Understanding inverse functions and their graphical representation is a fundamental concept in mathematics, particularly in calculus. By following the steps outlined in this article, you can confidently find and verify inverse functions for a wide range of problems. Keep practicing, and you'll become a pro at this in no time!