Are fx 3x 7 and gx 1/3x - 7 Inverse Functions? A Comprehensive Analysis

The concept of inverse functions is fundamental in mathematics, particularly in algebra and calculus. To determine whether two given functions, fx and gx, are inverse functions, one must check if the composition of the functions in both orders results in the identity function, i.e., fgx x and gfx x. This article provides a detailed analysis of the given functions fx 3x 7 and gx 1/3x - 7 to explore whether these functions are indeed inverse functions.

Introduction to Inverse Functions

Two functions, fx and gx, are considered inverse functions if they satisfy the following conditions:

fgx x gfx x

This means that applying one function and then the other should result in the original input value. Inverse functions are particularly useful in solving equations and understanding the relationships in mathematical models.

Given Functions and Objectives

Let's consider the given functions:

fx 3x 7

gx 1/3x - 7

The objective is to determine if these functions are inverse functions by checking if the composition of these functions, in both orders, results in the identity function x.

Step-by-Step Analysis

Step 1: Compute fogx

First, we need to substitute gx into fx:

Calculate fogx: Substitute gx into fx:

Let's proceed with the calculations:

fgx  f(1/3x - 7)  3(1/3x - 7)   7

Distributing the 3:

3(1/3x - 7) 7 x - 21 7

Combining like terms:

x - 14

Since we have:

f ( 1 3 x - 7 ) x - 14

This shows that fogx ≠ x, so we need to check the other composition, gfx.

Step 2: Compute gfx

We now substitute fx into gx:

Calculate gfx: Substitute fx into gx:

Let's perform the computation:

gfx  g(3x   7)  1/3 * (3x   7) - 7

Distributing 1/3:

1/3 * (3x 7) - 7 x 7/3 - 7

Converting 7 as a fraction:

x 7/3 - 21/3 x - 14/3

Since we have:

g ( 3 x 7 ) 1 3 ( 3 x 7 ) - 7 x - 14 3

Since neither fogx x nor gfx x holds true, the given functions fx 3x 7 and gx 1/3x - 7 are not inverse functions.

Conclusion

In conclusion, after thorough analysis, it has been determined that the given functions do not satisfy the conditions for being inverse functions. Understanding and correctly applying the rules of function composition and algebraic manipulation are crucial in validating whether two functions are indeed inverses of each other.

If you have further questions or need more clarification, feel free to ask. The importance of precision in mathematical notation also cannot be overstated, especially in distinguishing between expressions like 1/3x - 7 and 1/3x - 7.