Domain and Range of a Quadratic Function

Oct 13, 2025 by thesweetdesigns.com 33 views

Introduction

In mathematics, functions are used to describe the relationship between variables. A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The function $f(x) = x^2 - 4$ is a quadratic function that can be used to model various real-world situations. In this article, we will explore the domain and range of the function $f(x) = x^2 - 4$ and find its inverse function.

Domain and Range of a Quadratic Function

The domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of $x$ for which the function $f(x)$ is defined. The range of a function is the set of all possible output values for which the function is defined.

For the function $f(x) = x^2 - 4$ , we can see that the function is defined for all real numbers, i.e., $x \in (-\infty, \infty)$ . This is because the square of any real number is always non-negative, and subtracting 4 from a non-negative number will always result in a non-positive number.

Therefore, the domain of the function $f(x) = x^2 - 4$ is $(-\infty, \infty)$ .

The range of a function is the set of all possible output values for which the function is defined. For the function $f(x) = x^2 - 4$ , we can see that the function will always produce non-positive values, i.e., $f(x) \leq 0$ . This is because the square of any real number is always non-negative, and subtracting 4 from a non-negative number will always result in a non-positive number.

Therefore, the range of the function $f(x) = x^2 - 4$ is $(-\infty, \infty)$ .

However, this is incorrect. The correct range is [-4, ∞).

Finding the Inverse Function

The inverse function of a function $f(x)$ is denoted by $f^{-1}(x)$ and is defined as the function that undoes the action of the original function. In other words, if $f(x)$ maps an input $x$ to an output $y$ , then the inverse function $f^{-1}(x)$ maps the output $y$ back to the input $x$ .

To find the inverse function of $f(x) = x^2 - 4$ , we can start by writing the function as $y = x^2 - 4$ . Then, we can interchange the roles of $x$ and $y$ to get $x = y^2 - 4$ .

Next, we can solve for $y$ by adding 4 to both sides of the equation to get $x + 4 = y^2$ . Then, we can take the square root of both sides of the equation to get $y = \pm \sqrt{x + 4}$ .

Therefore, the inverse function of $f(x) = x^2 - 4$ is $f^{-1}(x) = \pm \sqrt{x + 4}$ .

Domain and Range of the Inverse Function

The domain of the inverse function is the set of all possible input values for which the inverse function is defined. In other words, it is the set of all possible values of $x$ for which the inverse function $f^{-1}(x)$ is defined.

For the inverse function $f^{-1}(x) = \pm \sqrt{x + 4}$ , we can see that the function is defined for all real numbers, i.e., $x \in (-\infty, \infty)$ . This is because the square root of any non-negative real number is always defined.

Therefore, the domain of the inverse function $f^{-1}(x) = \pm \sqrt{x + 4}$ is $(-\infty, \infty)$ .

The range of the inverse function is the set of all possible output values for which the inverse function is defined. For the inverse function $f^{-1}(x) = \pm \sqrt{x + 4}$ , we can see that the function will always produce non-negative values, i.e., $f^{-1}(x) \geq 0$ .

Therefore, the range of the inverse function $f^{-1}(x) = \pm \sqrt{x + 4}$ is $[0, \infty)$ .

However, this is incorrect. The correct range is [-4, ∞).

Conclusion

In this article, we have explored the domain and range of the function $f(x) = x^2 - 4$ and found its inverse function. We have also discussed the domain and range of the inverse function.

The domain of the function $f(x) = x^2 - 4$ is $(-\infty, \infty)$ , and the range is $[-4, \infty)$ .

The inverse function of $f(x) = x^2 - 4$ is $f^{-1}(x) = \pm \sqrt{x + 4}$ , and the domain and range of the inverse function are $(-\infty, \infty)$ and $[-4, \infty)$ , respectively.

References

[1] "Quadratic Functions" by Math Open Reference
[2] "Inverse Functions" by Khan Academy

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about quadratic functions, including the domain and range of a quadratic function, finding the inverse of a quadratic function, and more.

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. It can be written in the form $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants.

Q: What is the domain of a quadratic function?

A: The domain of a quadratic function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of $x$ for which the function $f(x)$ is defined. For a quadratic function, the domain is always $(-\infty, \infty)$ .

Q: What is the range of a quadratic function?

A: The range of a quadratic function is the set of all possible output values for which the function is defined. In other words, it is the set of all possible values of $f(x)$ for which the function is defined. The range of a quadratic function can be any set of real numbers, depending on the specific function.

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

A: To find the inverse of a quadratic function, you need to interchange the roles of $x$ and $y$ and then solve for $y$ . This will give you the inverse function.

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

A: The domain of the inverse of a quadratic function is the set of all possible input values for which the inverse function is defined. In other words, it is the set of all possible values of $x$ for which the inverse function $f^{-1}(x)$ is defined. The range of the inverse of a quadratic function is the set of all possible output values for which the inverse function is defined.

Q: Can I use a quadratic function to model real-world situations?

A: Yes, quadratic functions can be used to model many real-world situations, such as the trajectory of a projectile, the motion of an object under the influence of gravity, and the growth of a population.

Q: What are some common applications of quadratic functions?

A: Some common applications of quadratic functions include:

Modeling the trajectory of a projectile
Modeling the motion of an object under the influence of gravity
Modeling the growth of a population
Modeling the cost of producing a product
Modeling the revenue of a business

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you need to plot the points on the graph and then connect them with a smooth curve. You can also use a graphing calculator or software to graph the function.

Q: What are some common mistakes to avoid when working with quadratic functions?

A: Some common mistakes to avoid when working with quadratic functions include:

Not checking the domain and range of the function
Not using the correct formula for the inverse function
Not checking the validity of the solutions
Not using the correct method for graphing the function

Conclusion

In this article, we have answered some of the most frequently asked questions about quadratic functions, including the domain and range of a quadratic function, finding the inverse of a quadratic function, and more. We hope that this article has been helpful in understanding quadratic functions and their applications.

References

[1] "Quadratic Functions" by Math Open Reference
[2] "Inverse Functions" by Khan Academy
[3] "Graphing Quadratic Functions" by Wolfram MathWorld