Introduction

When dealing with functions in mathematics, particularly inverse functions, it is important to understand the relationships between original functions and their inverses. In this article, we will explore the inverse of the function

g(x) ln(x) - 3

Let's embark on this mathematical journey step-by-step.

Step 1: Understanding the Function

The given function is

g(x) ln(x) - 3

The function involves the natural logarithm (ln), which is the logarithm to the base e, where e is approximately 2.71828.

Step 2: Definition of Inverse Functions

An inverse function, denoted as g-1(x), is a function that "undoes" the original function g(x). Mathematically, this can be expressed as:

g(g-1(x)) x

Using this definition, we can find the inverse of our given function.

Step 3: Finding the Inverse Function

Let's denote the inverse function as y g-1(x).

Starting from the original function:

g(x) ln(x) - 3

We substitute x with y and y with x to find the inverse:

y ln(x) - 3

Now, solve for x:

Rearrange the equation to isolate ln(x>:ln(x) y 3Exponentiate both sides with base e:x ey 3Substitute y back with x to represent the inverse function:g-1(x) ex 3

Thus, the inverse function is:

g-1(x) ex 3

Step 4: Verification

To verify, let's substitute g-1(x) into the original function:

g(g-1(x)) g(ex 3)

Simplify using the definition of g(x):

g(ex 3) ln(ex 3) - 3

Using the property of logarithms ln(ez) z, we get:

g(ex 3) x 3 - 3 x

This confirms that g-1(x) ex 3 is indeed the inverse function.

Conclusion

Understanding how to find the inverse of a function, particularly one involving logarithms, is a crucial concept in advanced mathematics and its applications. In this article, we have explored finding the inverse of a logarithmic function, g(x) ln(x) - 3, and derived the inverse function as g-1(x) ex 3.

If you have any further questions or need further clarification, feel free to ask.