Finding the roots of a function

Oct 13, 2025 by miodeya.com 102 views

What is the Inverse of the Function f(x) = 2x - 10?

Understanding the Concept of Inverse Functions

In mathematics, an inverse function is a function that reverses the operation of another function. In other words, if we have a function f(x) that maps an input x to an output f(x), then the inverse function h(x) maps the output f(x) back to the input x. Inverse functions are denoted by the notation f^(-1)(x) or h(x).

The Given Function f(x) = 2x - 10

The given function is f(x) = 2x - 10. This is a linear function that takes an input x and produces an output f(x) = 2x - 10. To find the inverse of this function, we need to swap the roles of x and f(x) and solve for f(x).

Step 1: Swap the Roles of x and f(x)

We start by swapping the roles of x and f(x) to get x = 2f(x) - 10.

Step 2: Solve for f(x)

Now, we need to solve for f(x). We can do this by isolating f(x) on one side of the equation.

First, we add 10 to both sides of the equation to get x + 10 = 2f(x).

Next, we divide both sides of the equation by 2 to get (x + 10)/2 = f(x).

The Inverse Function h(x)

Therefore, the inverse function h(x) is given by h(x) = (x + 10)/2.

Simplifying the Inverse Function

We can simplify the inverse function by multiplying both the numerator and the denominator by 1/2 to get h(x) = (x + 10)/2 * (1/2) = (x + 10)/4.

Comparing with the Options

Now, let's compare our answer with the options given:

A. h(x) = 2x - 5 B. h(x) = 2x + 5 C. h(x) = (1/2)x - 5 D. h(x) = (1/2)x + 5

Our answer is h(x) = (x + 10)/4, which is not among the options. However, we can simplify it further to get h(x) = (1/4)x + 5/2.

Conclusion

Therefore, the correct answer is not among the options given. However, we can see that option C is close to our answer, but it is not exactly the same. The correct answer is h(x) = (1/4)x + 5/2.

Understanding the Concept of Inverse Functions

Inverse functions are an important concept in mathematics, and they have many applications in various fields such as physics, engineering, and economics. Inverse functions are used to solve equations, find the roots of a function, and analyze the behavior of a function.

Properties of Inverse Functions

Inverse functions have several properties that make them useful in mathematics. Some of the properties of inverse functions include:

One-to-one correspondence: An inverse function is a one-to-one correspondence between the input and output of a function.
Symmetry: An inverse function is symmetric about the line y = x.
Reversibility: An inverse function is reversible, meaning that if we apply the inverse function to the output of the original function, we get back the original input.

Applications of Inverse Functions

Inverse functions have many applications in various fields such as physics, engineering, and economics. Some of the applications of inverse functions include:

Solving equations: Inverse functions are used to solve equations by finding the roots of a function.
Finding the roots of a function: Inverse functions are used to find the roots of a function by applying the inverse function to the output of the original function.
Analyzing the behavior of a function: Inverse functions are used to analyze the behavior of a function by applying the inverse function to the output of the original function.

Conclusion

In conclusion, inverse functions are an important concept in mathematics, and they have many applications in various fields such as physics, engineering, and economics. Inverse functions are used to solve equations, find the roots of a function, and analyze the behavior of a function.
Inverse Functions: A Q&A Guide

Understanding Inverse Functions

Inverse functions are a fundamental concept in mathematics, and they have many applications in various fields such as physics, engineering, and economics. In this article, we will provide a comprehensive guide to inverse functions, including their definition, properties, and applications.

Q: What is an inverse function?

A: An inverse function is a function that reverses the operation of another function. In other words, if we have a function f(x) that maps an input x to an output f(x), then the inverse function h(x) maps the output f(x) back to the input x.

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

A: To find the inverse of a function, you need to swap the roles of x and f(x) and solve for f(x). This involves reversing the operation of the original function.

Q: What are the properties of inverse functions?

A: Inverse functions have several properties that make them useful in mathematics. Some of the properties of inverse functions include:

One-to-one correspondence: An inverse function is a one-to-one correspondence between the input and output of a function.
Symmetry: An inverse function is symmetric about the line y = x.
Reversibility: An inverse function is reversible, meaning that if we apply the inverse function to the output of the original function, we get back the original input.

Q: What are the applications of inverse functions?

A: Inverse functions have many applications in various fields such as physics, engineering, and economics. Some of the applications of inverse functions include:

Solving equations: Inverse functions are used to solve equations by finding the roots of a function.
Finding the roots of a function: Inverse functions are used to find the roots of a function by applying the inverse function to the output of the original function.
Analyzing the behavior of a function: Inverse functions are used to analyze the behavior of a function by applying the inverse function to the output of the original function.

Q: How do I determine if a function is invertible?

A: A function is invertible if it is one-to-one, meaning that each output value corresponds to exactly one input value. In other words, a function is invertible if it passes the horizontal line test.

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

A: An inverse function is a function that reverses the operation of another function, while a reciprocal function is a function that takes the reciprocal of the input value. For example, the inverse of the function f(x) = 2x - 10 is h(x) = (x + 10)/2, while the reciprocal of the function f(x) = 2x - 10 is f(x) = 1/(2x - 10).

Q: Can I have multiple inverse functions?

A: Yes, it is possible to have multiple inverse functions. For example, the function f(x) = x^2 has two inverse functions: h(x) = sqrt(x) and h(x) = -sqrt(x).

Q: How do I graph an inverse function?

A: To graph an inverse function, you need to reflect the graph of the original function about the line y = x. This involves swapping the x and y coordinates of each point on the graph.

Conclusion

In conclusion, inverse functions are an important concept in mathematics, and they have many applications in various fields such as physics, engineering, and economics. In this article, we have provided a comprehensive guide to inverse functions, including their definition, properties, and applications. We hope that this guide has been helpful in understanding inverse functions and their importance in mathematics.

Common Mistakes to Avoid

When working with inverse functions, there are several common mistakes to avoid. Some of these mistakes include:

Confusing the inverse function with the reciprocal function: Make sure to distinguish between the inverse function and the reciprocal function.
Not checking for one-to-one correspondence: Make sure to check if the function is one-to-one before finding its inverse.
Not reflecting the graph about the line y = x: Make sure to reflect the graph of the original function about the line y = x when graphing the inverse function.

Real-World Applications

Inverse functions have many real-world applications in various fields such as physics, engineering, and economics. Some of the real-world applications of inverse functions include:

Solving equations: Inverse functions are used to solve equations by finding the roots of a function.
Finding the roots of a function: Inverse functions are used to find the roots of a function by applying the inverse function to the output of the original function.
Analyzing the behavior of a function: Inverse functions are used to analyze the behavior of a function by applying the inverse function to the output of the original function.

Conclusion