What is the Inverse of y √x?

Understanding and finding the inverse of a function, particularly the inverse of the square root function, is a fundamental concept in mathematics. This article guides you through the steps to find the inverse of the function y √x and explains the significance of the domain in ensuring that the inverse is valid.

Understanding the Original Function

The function y √x, where x ge; 0, describes a relationship where y is the non-negative square root of x. This function is defined only for non-negative values of x because the square root of a negative number would yield an imaginary result, which is outside the scope of real numbers.

Steps to Find the Inverse Function

To find the inverse of a function, we follow a systematic approach:

1. Rewrite the Original Equation

The starting point is the original equation:

y √x

2. Swap x and y

Interchanging x and y is the first step in finding the inverse:

x √y

3. Solve for y

To solve for y, we square both sides of the equation to eliminate the square root:

x2 y

4. Write the Inverse Function

The inverse function is:

y x2

It is essential to note that this inverse function is valid only when x ge; 0. This is because the domain of the original function y √x is defined for non-negative x, and the range of the inverse function is also non-negative.

Therefore, the inverse function of y √x is:

y x2 (for x ge; 0)

Alternative Methods for Finding the Inverse

There are alternative methods to find the inverse of the function y √x using logarithms:

1. Using Logarithms

The square root function can be expressed as a power of 0.5:

y x1/2

Interchanging x and y:

x y1/2

To solve for y, we take the logarithm of both sides:

log(x) (1/2)log(y)

Multiplying both sides by 2:

2log(x) log(y)

Therefore:

y x2 (where x ge; 0 and y ge; 0)

Conclusion

The process of finding the inverse of the function y √x involves several key mathematical concepts, including swapping variables and solving equations. Additionally, understanding the domain and range of the original and inverse functions is crucial for ensuring the validity and applicability of the inverse.

By following these steps, you can find the inverse of the function y √x and apply the concept to other functions. This article has provided a detailed and comprehensive guide to help you understand and solve inverse functions effectively.