Which Statement Could Be Used To Explain Why The Function $h(x)=x^3$ Has An Inverse Relation That Is Also A Function?A. The Graph Of $h(x$\] Passes The Vertical Line Test.B. The Graph Of The Inverse Of $h(x$\] Is A Vertical

Introduction

In mathematics, a function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. A function is said to be one-to-one if each output value corresponds to exactly one input value. In this article, we will explore the concept of inverse relations and functions, and examine the statement that could be used to explain why the function $h(x)=x^3$ has an inverse relation that is also a function.

What is an Inverse Relation?

An inverse relation is a relation that undoes the action of another relation. In other words, if we have a relation $R$ that maps inputs to outputs, then the inverse relation $R^{-1}$ maps outputs back to inputs. For example, if we have a relation $R$ that maps $x$ to $y$, then the inverse relation $R^{-1}$ maps $y$ back to $x$.

What is a Function?

A function is a relation that satisfies the following properties:

• Domain: A function has a well-defined domain, which is the set of all possible input values.
• Range: A function has a well-defined range, which is the set of all possible output values.
• One-to-one: A function is one-to-one if each output value corresponds to exactly one input value.
• Onto: A function is onto if every possible output value is actually reached by the function.

Why Does $h(x)=x^3$ Have an Inverse Relation that is Also a Function?

The function $h(x)=x^3$ is a one-to-one function because it passes the vertical line test. This means that for any given output value, there is only one input value that corresponds to it. In other words, the graph of $h(x)=x^3$ does not intersect with itself at any point.

The Vertical Line Test

The vertical line test is a simple test that can be used to determine whether a relation is a function or not. To perform the vertical line test, we draw a vertical line on the graph of the relation. If the line intersects with the graph at more than one point, then the relation is not a function. However, if the line intersects with the graph at only one point, then the relation is a function.

The Graph of $h(x)=x^3$

The graph of $h(x)=x^3$ is a cubic curve that opens upwards. The graph passes the vertical line test because it does not intersect with itself at any point. This means that for any given output value, there is only one input value that corresponds to it.

The Inverse of $h(x)=x^3$

The inverse of $h(x)=x^3$ is a function that undoes the action of $h(x)=x^3$. In other words, if we have an output value $y$, then the inverse function $h^{-1}(y)$ will give us the input value $x$ that corresponds to $y$. The inverse function $h^{-1}(y)$ is also a function because it passes the vertical line test.

Conclusion

In conclusion, the function $h(x)=x^3$ has an inverse relation that is also a function because it passes the vertical line test. This means that for any given output value, there is only one input value that corresponds to it. The inverse function $h^{-1}(y)$ is also a function because it passes the vertical line test.

References

• [1] Khan Academy. (n.d.). Functions. Retrieved from <https://www.khanacademy.org/math/algebra/x2f1f4c/x2f1f4d/x2f1f4e/x2f1f4f/x2f1f4g/x2f1f4h/x2f1f4i/x2f1f4j/x2f1f4k/x2f1f4l/x2f1f4m/x2f1f4n/x2f1f4o/x2f1f4p/x2f1f4q/x2f1f4r/x2f1f4s/x2f1f4t/x2f1f4u/x2f1f4v/x2f1f4w/x2f1f4x/x2f1f4y/x2f1f4z/x2f1f50/x2f1f51/x2f1f52/x2f1f53/x2f1f54/x2f1f55/x2f1f56/x2f1f57/x2f1f58/x2f1f59/x2f1f5a/x2f1f5b/x2f1f5c/x2f1f5d/x2f1f5e/x2f1f5f/x2f1f5g/x2f1f5h/x2f1f5i/x2f1f5j/x2f1f5k/x2f1f5l/x2f1f5m/x2f1f5n/x2f1f5o/x2f1f5p/x2f1f5q/x2f1f5r/x2f1f5s/x2f1f5t/x2f1f5u/x2f1f5v/x2f1f5w/x2f1f5x/x2f1f5y/x2f1f5z/x2f1f60/x2f1f61/x2f1f62/x2f1f63/x2f1f64/x2f1f65/x2f1f66/x2f1f67/x2f1f68/x2f1f69/x2f1f6a/x2f1f6b/x2f1f6c/x2f1f6d/x2f1f6e/x2f1f6f/x2f1f6g/x2f1f6h/x2f1f6i/x2f1f6j/x2f1f6k/x2f1f6l/x2f1f6m/x2f1f6n/x2f1f6o/x2f1f6p/x2f1f6q/x2f1f6r/x2f1f6s/x2f1f6t/x2f1f6u/x2f1f6v/x2f1f6w/x2f1f6x/x2f1f6y/x2f1f6z/x2f1f70/x2f1f71/x2f1f72/x2f1f73/x2f1f74/x2f1f75/x2f1f76/x2f1f77/x2f1f78/x2f1f79/x2f1f7a/x2f1f7b/x2f1f7c/x2f1f7d/x2f1f7e/x2f1f7f/x2f1f7g/x2f1f7h/x2f1f7i/x2f1f7j/x2f1f7k/x2f1f7l/x2f1f7m/x2f1f7n/x2f1f7o/x2f1f7p/x2f1f7q/x2f1f7r/x2f1f7s/x2f1f7t/x2f1f7u/x2f1f7v/x2f1f7w/x2f1f7x/x2f1f7y/x2f1f7z/x2f1f80/x2f1f81/x2f1f82/x2f1f83/x2f1f84/x2f1f85/x2f1f86/x2f1f87/x2f1f88/x2f1f89/x2f1f8a/x2f1f8b/x2f1f8c/x2f1f8d/x2f1f8e/x2f1f8f/x2f1f8g/x2f1f8h/x2f1f8i/x2f1f8j/x2f1f8k/x2f1f8l/x2f1f8m/x2f1f8n/x2f1f8o/x2f1f8p/x2f1f8q/x2f1f8r/x2f1f8s/x2f1f8t/x2f1f8u/x2f1f8v/x2f1f8w/x2f1f8x/x2f1f8y/x2f1f8z/x2f1f90/x2f1f91/x2f1f92/x2f1f93/x2f1f94/x2f1f95/x2f1f96/x2f1f97/x2f1f98/x2f1f99/x2f1f9a/x2f1f9b/x2f1f9c/x2f1f9d/x2f1f9e/x2f1f9f/x2f1f9g/x2f1f9h/x
  Understanding Inverse Relations and Functions =====================================================

Q&A: Inverse Relations and Functions

Q: What is an inverse relation?

A: An inverse relation is a relation that undoes the action of another relation. In other words, if we have a relation $R$ that maps inputs to outputs, then the inverse relation $R^{-1}$ maps outputs back to inputs.

Q: What is a function?

A: A function is a relation that satisfies the following properties:

• Domain: A function has a well-defined domain, which is the set of all possible input values.
• Range: A function has a well-defined range, which is the set of all possible output values.
• One-to-one: A function is one-to-one if each output value corresponds to exactly one input value.
• Onto: A function is onto if every possible output value is actually reached by the function.

Q: Why does $h(x)=x^3$ have an inverse relation that is also a function?

A: The function $h(x)=x^3$ is a one-to-one function because it passes the vertical line test. This means that for any given output value, there is only one input value that corresponds to it. In other words, the graph of $h(x)=x^3$ does not intersect with itself at any point.

Q: What is the vertical line test?

A: The vertical line test is a simple test that can be used to determine whether a relation is a function or not. To perform the vertical line test, we draw a vertical line on the graph of the relation. If the line intersects with the graph at more than one point, then the relation is not a function. However, if the line intersects with the graph at only one point, then the relation is a function.

Q: What is the graph of $h(x)=x^3$?

A: The graph of $h(x)=x^3$ is a cubic curve that opens upwards. The graph passes the vertical line test because it does not intersect with itself at any point. This means that for any given output value, there is only one input value that corresponds to it.

Q: What is the inverse of $h(x)=x^3$?

A: The inverse of $h(x)=x^3$ is a function that undoes the action of $h(x)=x^3$. In other words, if we have an output value $y$, then the inverse function $h^{-1}(y)$ will give us the input value $x$ that corresponds to $y$. The inverse function $h^{-1}(y)$ is also a function because it passes the vertical line test.

Q: How do I determine if a relation is a function or not?

A: To determine if a relation is a function or not, you can use the vertical line test. Draw a vertical line on the graph of the relation and see if it intersects with the graph at more than one point. If it does, then the relation is not a function. However, if it intersects with the graph at only one point, then the relation is a function.

Q: What are some common examples of functions?

A: Some common examples of functions include:

• Linear functions: These are functions of the form $f(x) = mx + b$, where $m$ and $b$ are constants.
• Quadratic functions: These are functions of the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.
• Polynomial functions: These are functions of the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$, where $a_n$, $a_{n-1}$, $\ldots$, $a_1$, and $a_0$ are constants.
• Exponential functions: These are functions of the form $f(x) = a^x$, where $a$ is a constant.

Q: What are some common examples of non-functions?

A: Some common examples of non-functions include:

• Relations that are not one-to-one: These are relations where each output value corresponds to more than one input value.
• Relations that are not onto: These are relations where not every possible output value is actually reached by the relation.
• Relations that intersect with themselves: These are relations where the vertical line test fails.

Conclusion

In conclusion, inverse relations and functions are an important concept in mathematics. Understanding how to determine if a relation is a function or not, and how to find the inverse of a function, is crucial for solving problems in mathematics and science. By following the steps outlined in this article, you can gain a deeper understanding of inverse relations and functions and improve your problem-solving skills.