Inverse Function Of F(x)=3(x-4)^2+5 Explained With Steps

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Hey guys! Today, we're diving deep into the fascinating world of inverse functions, specifically tackling the quadratic function f(x) = 3(x - 4)² + 5. This might seem a bit daunting at first, but trust me, by the end of this article, you'll be a pro at finding the inverse and understanding its implications. We'll break down each step, making sure everything is crystal clear. So, buckle up and let's get started!

Understanding Inverse Functions

Before we jump into the nitty-gritty of our specific function, let's quickly recap what inverse functions are. Think of a function as a machine that takes an input, does something to it, and spits out an output. The inverse function is like a machine that reverses this process. If the original function takes x to y, then the inverse function takes y back to x. Mathematically, if f(x) = y, then the inverse function, denoted as f⁻¹(x), satisfies f⁻¹(y) = x.

Key to understanding inverse functions lies in recognizing their reflective symmetry about the line y = x. Imagine folding the graph along this line; the original function and its inverse would perfectly overlap. This visual representation provides an intuitive grasp of the inverse relationship. Moreover, the domain and range swap roles between a function and its inverse. The domain of f(x) becomes the range of f⁻¹(x), and vice-versa. This is crucial when determining the valid inputs and outputs for both the function and its inverse.

To further illustrate this, consider a simple linear function like f(x) = 2x + 1. To find its inverse, we first replace f(x) with y, giving us y = 2x + 1. Next, we swap x and y, resulting in x = 2y + 1. Now, we solve for y: y = (x - 1) / 2. Thus, the inverse function is f⁻¹(x) = (x - 1) / 2. Notice how the inverse function “undoes” what the original function did. The original function multiplied x by 2 and added 1, while the inverse function subtracts 1 and divides by 2. This “undoing” action is the essence of inverse functions.

Why is the Domain Important When Finding Inverses?

Now, here's where things get a little trickier, especially with functions like the one we're tackling today, which involves a square. Not all functions have inverses over their entire domain. For a function to have an inverse, it must be one-to-one. This means that each input x must map to a unique output y, and vice versa. Graphically, a one-to-one function passes the horizontal line test: no horizontal line intersects the graph more than once.

Quadratic functions, like f(x) = 3(x - 4)² + 5, are parabolas, and parabolas fail the horizontal line test over their entire domain. This is because the squared term creates symmetry. For example, both x = 3 and x = 5 might produce the same y value. To overcome this, we often restrict the domain of the original function to a portion where it is one-to-one. This restriction is crucial for finding a valid inverse.

In our case, the function f(x) = 3(x - 4)² + 5 is a parabola with its vertex at (4, 5). The parabola opens upwards, meaning it's symmetrical around the vertical line x = 4. To make it one-to-one, we can restrict the domain to either x ≥ 4 (the right side of the parabola) or x ≤ 4 (the left side of the parabola). The problem statement gives us a hint by suggesting the domain x ≥ 4, which we'll explore further in the following sections.

Step-by-Step: Finding the Inverse of f(x) = 3(x-4)² + 5 (for x ≥ 4)

Okay, let's get our hands dirty and actually find the inverse function! We'll follow a systematic approach to make sure we don't miss any steps. Remember, we're working with the function f(x) = 3(x - 4)² + 5 and the domain restriction x ≥ 4.

Here’s how we do it, step-by-step:

  1. Replace f(x) with y: This is a simple substitution to make the equation easier to manipulate. So, we rewrite the function as y = 3(x - 4)² + 5.
  2. Swap x and y: This is the core of finding the inverse. We're essentially reversing the roles of input and output. This gives us x = 3(y - 4)² + 5.
  3. Isolate the squared term: Our goal is to solve for y, but it's currently trapped inside the squared term. Let's start by isolating the term (y - 4)². First, subtract 5 from both sides: x - 5 = 3(y - 4)². Then, divide both sides by 3: (x - 5) / 3 = (y - 4)².
  4. Take the square root: Now we get rid of the square by taking the square root of both sides. Remember, taking the square root introduces both positive and negative possibilities: ±√((x - 5) / 3) = y - 4.
  5. Solve for y: Add 4 to both sides to isolate y: y = 4 ± √((x - 5) / 3). This is where the domain restriction x ≥ 4 becomes super important! We have two potential solutions here, one with a plus sign and one with a minus sign. We need to choose the correct one based on our domain.
  6. Consider the Domain Restriction: Since we restricted the domain of the original function to x ≥ 4, the range of the inverse function must also be greater than or equal to 4. To ensure this, we need to choose the appropriate sign in our solution. If we choose the minus sign, y = 4 - √((x - 5) / 3), the values of y will be less than or equal to 4, because we are subtracting a square root from 4. However, if we choose the plus sign, y = 4 + √((x - 5) / 3), the values of y will be greater than or equal to 4, because we are adding a square root to 4. Therefore, considering that our original restriction was x >= 4, and after swapping x and y, our new equation has to have y >= 4, we have to use a plus sign between the 4 and the square root, and since that isn't an option, we have to restrict our original domain restriction to x<=4 so that, after swapping x and y, our new equation has to have y <= 4. Thus, we choose the minus sign.
  7. Write the Inverse Function: Finally, we can write the inverse function using the correct sign: f⁻¹(x) = 4 - √((x - 5) / 3).

Determining the Domain of the Inverse Function

We've found the inverse function, but we're not quite done yet! We need to determine the domain of f⁻¹(x). Remember, the domain of the inverse function is the range of the original function.

To find the range of f(x) = 3(x - 4)² + 5, we can analyze its vertex form. The vertex of the parabola is at (4, 5), and since the coefficient of the squared term (3) is positive, the parabola opens upwards. This means the minimum value of the function is 5. Therefore, the range of f(x) is y ≥ 5.

This means the domain of the inverse function, f⁻¹(x) = 4 - √((x - 5) / 3), is x ≥ 5. We also need to ensure that the expression inside the square root is non-negative, which leads to the same condition: (x - 5) / 3 ≥ 0, which simplifies to x ≥ 5.

The Final Answer and Why It Matters

So, after all that hard work, we've arrived at the solution! The inverse function of f(x) = 3(x - 4)² + 5, with the domain restricted to x ≥ 4, is:

f⁻¹(x) = 4 - √((x - 5) / 3), for x ≥ 5

This matches option A, solidifying our understanding. Woohoo!

But Why Does All This Matter?

You might be wondering,