Consider The Following Function On The Given Domain: A ( X ) = ( X − 4 ) 2 + 4 , X ≥ 4 A(x) = (x-4)^2 + 4, \quad X \geq 4 A ( X ) = ( X − 4 ) 2 + 4 , X ≥ 4 Step 1 Of 2: Find A Formula For The Inverse Of The Function On The Given Domain, If Possible.

Introduction

In this article, we will explore the concept of finding the inverse of a quadratic function on a given domain. The function we will be working with is $A(x) = (x-4)^2 + 4$, where $x \geq 4$. We will follow a step-by-step approach to find the inverse of this function, if possible.

Step 1: Understanding the Function

The given function is a quadratic function in the form of $f(x) = a(x-h)^2 + k$, where $a$, $h$, and $k$ are constants. In this case, $a = 1$, $h = 4$, and $k = 4$. The function is defined for $x \geq 4$, which means that the domain of the function is $[4, \infty)$.

Step 2: Finding the Inverse

To find the inverse of a function, we need to swap the roles of $x$ and $y$ and then solve for $y$. Let's start by writing the function as $y = (x-4)^2 + 4$. To find the inverse, we will swap $x$ and $y$ to get $x = (y-4)^2 + 4$.

Solving for y

Now, we need to solve for $y$. We can start by subtracting $4$ from both sides of the equation to get $x - 4 = (y-4)^2$. Next, we can take the square root of both sides to get $\sqrt{x - 4} = y - 4$. Finally, we can add $4$ to both sides to get $y = \sqrt{x - 4} + 4$.

Domain of the Inverse

Since the original function is defined for $x \geq 4$, the domain of the inverse function will be $[4, \infty)$. However, we need to check if the inverse function is defined for this domain.

Checking the Inverse

Let's check if the inverse function is defined for $x \geq 4$. We can plug in a value of $x$ into the inverse function to see if it is defined. For example, let's plug in $x = 5$. We get $y = \sqrt{5 - 4} + 4 = \sqrt{1} + 4 = 5$. This shows that the inverse function is defined for $x \geq 4$.

Conclusion

In this article, we found the inverse of the quadratic function $A(x) = (x-4)^2 + 4$ on the given domain $x \geq 4$. The inverse function is $y = \sqrt{x - 4} + 4$. We also checked that the inverse function is defined for the domain $[4, \infty)$.

Example

Let's consider an example to illustrate the concept of finding the inverse of a quadratic function. Suppose we have a function $f(x) = x^2 + 2x + 1$. We can find the inverse of this function by following the same steps as before.

Step 1: Understanding the Function

The function $f(x) = x^2 + 2x + 1$ is a quadratic function in the form of $f(x) = ax^2 + bx + c$, where $a = 1$, $b = 2$, and $c = 1$. The domain of the function is $(-\infty, \infty)$.

Step 2: Finding the Inverse

To find the inverse of the function, we need to swap the roles of $x$ and $y$ and then solve for $y$. Let's start by writing the function as $y = x^2 + 2x + 1$. To find the inverse, we will swap $x$ and $y$ to get $x = y^2 + 2y + 1$.

Solving for y

Now, we need to solve for $y$. We can start by subtracting $1$ from both sides of the equation to get $x - 1 = y^2 + 2y$. Next, we can rearrange the equation to get $y^2 + 2y + (x - 1) = 0$. We can then use the quadratic formula to solve for $y$.

Quadratic Formula

The quadratic formula is given by $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. In this case, $a = 1$, $b = 2$, and $c = x - 1$. Plugging these values into the quadratic formula, we get $y = \frac{-2 \pm \sqrt{2^2 - 4(1)(x - 1)}}{2(1)}$.

Simplifying the Expression

Simplifying the expression, we get $y = \frac{-2 \pm \sqrt{4 - 4x + 4}}{2} = \frac{-2 \pm \sqrt{-4x + 8}}{2} = \frac{-2 \pm 2\sqrt{-x + 2}}{2} = -1 \pm \sqrt{-x + 2}$.

Domain of the Inverse

Since the original function is defined for $x \in (-\infty, \infty)$, the domain of the inverse function will be $(-\infty, 2)$.

Checking the Inverse

Let's check if the inverse function is defined for $x \in (-\infty, 2)$. We can plug in a value of $x$ into the inverse function to see if it is defined. For example, let's plug in $x = 0$. We get $y = -1 \pm \sqrt{-0 + 2} = -1 \pm \sqrt{2}$. This shows that the inverse function is defined for $x \in (-\infty, 2)$.

Conclusion

In this article, we found the inverse of the quadratic function $f(x) = x^2 + 2x + 1$ on the given domain $x \in (-\infty, \infty)$. The inverse function is $y = -1 \pm \sqrt{-x + 2}$. We also checked that the inverse function is defined for the domain $(-\infty, 2)$.

Conclusion

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Frequently Asked Questions

In this article, we will answer some frequently asked questions about finding the inverse of a quadratic function.

Q: What is the inverse of a quadratic function?

A: The inverse of a quadratic function is a function that undoes the action of the original function. In other words, if we have a function $f(x)$ and its inverse $f^{-1}(x)$, then $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

Q: How do I find the inverse of a quadratic function?

A: To find the inverse of a quadratic function, you need to follow these steps:

1. Write the function in the form $y = ax^2 + bx + c$.
2. Swap the roles of $x$ and $y$ to get $x = ay^2 + by + c$.
3. Solve for $y$ by rearranging the equation.
4. Check if the inverse function is defined for the given domain.

Q: What is the domain of the inverse of a quadratic function?

A: The domain of the inverse of a quadratic function is the set of all values of $x$ for which the original function is defined.

Q: Can I find the inverse of a quadratic function with a negative leading coefficient?

A: Yes, you can find the inverse of a quadratic function with a negative leading coefficient. However, the inverse function may not be defined for the entire domain of the original function.

Q: How do I check if the inverse function is defined for the given domain?

A: To check if the inverse function is defined for the given domain, you need to plug in a value of $x$ into the inverse function and see if it is defined. If the inverse function is defined for the given domain, then it is a valid inverse function.

Q: What is the difference between the inverse of a quadratic function and the reciprocal of a quadratic function?

A: The inverse of a quadratic function is a function that undoes the action of the original function, while the reciprocal of a quadratic function is a function that is the reciprocal of the original function. In other words, the inverse of a quadratic function is a function that "reverses" the original function, while the reciprocal of a quadratic function is a function that "flips" the original function.

Q: Can I find the inverse of a quadratic function with a complex coefficient?

A: Yes, you can find the inverse of a quadratic function with a complex coefficient. However, the inverse function may not be defined for the entire domain of the original function.

Q: How do I find the inverse of a quadratic function with a complex coefficient?

A: To find the inverse of a quadratic function with a complex coefficient, you need to follow the same steps as before. However, you may need to use complex numbers and complex arithmetic to solve for $y$.

Q: What is the significance of finding the inverse of a quadratic function?

A: Finding the inverse of a quadratic function is important in many areas of mathematics and science, such as algebra, calculus, and physics. It can help you solve equations, model real-world phenomena, and make predictions about the behavior of systems.

Conclusion

In conclusion, finding the inverse of a quadratic function involves swapping the roles of $x$ and $y$ and then solving for $y$. You can use the quadratic formula to solve for $y$ and then check if the inverse function is defined for the given domain. The inverse of a quadratic function is a function that undoes the action of the original function, and it can be used to solve equations, model real-world phenomena, and make predictions about the behavior of systems.