Problem 28 - Algebra for College Students

Question

Determine whether \(f\) and \(g\) are inverse functions. $$ f(x)=\frac{3}{4} x-2 \text { and } g(x)=\frac{4}{3} x+\frac{8}{3} $$

Step-by-Step Solution

Answer

The functions \( f \) and \( g \) are inverses.

Step 1: Define Inverses

Two functions \( f \) and \( g \) are inverse functions if \( f(g(x)) = x \) and \( g(f(x)) = x \) for all \( x \) in their domains.

Step 2: Calculate \( f(g(x)) \)

Substitute \( g(x) \) into \( f(x) \):\[f(g(x)) = f\left(\frac{4}{3}x + \frac{8}{3}\right) = \frac{3}{4}\left(\frac{4}{3}x + \frac{8}{3}\right) - 2\]Simplify:\[f(g(x)) = x + 2 - 2 = x\]Thus, \( f(g(x)) = x \).

Step 3: Calculate \( g(f(x)) \)

Substitute \( f(x) \) into \( g(x) \):\[g(f(x)) = g\left(\frac{3}{4}x - 2\right) = \frac{4}{3}\left(\frac{3}{4}x - 2\right) + \frac{8}{3}\]Simplify:\[g(f(x)) = x - \frac{8}{3} + \frac{8}{3} = x\]Thus, \( g(f(x)) = x \).

Step 4: Conclusion

Since both \( f(g(x)) = x \) and \( g(f(x)) = x \) are true, the functions \( f \) and \( g \) are inverses of each other.

Key Concepts

Function Composition

Function composition is an elegant mathematical concept that allows you to combine two functions to create a new function. Think of it as nesting a function within another. For two functions, say \( f(x) \) and \( g(x) \), you compose them as \( f(g(x)) \) or \( g(f(x)) \). This means you take the output of \( g(x) \) and input it into \( f(x) \), or vice versa.

In our exercise, we determined whether \( f(x) = \frac{3}{4}x - 2 \) and \( g(x) = \frac{4}{3}x + \frac{8}{3} \) are inverses by calculating both \( f(g(x)) \) and \( g(f(x)) \). If both result in the original input \( x \), then \( f \) and \( g \) are inverses of each other. So, essentially, function composition is about checking whether applying these functions one after another takes you back to your starting point.

Domain and Range

Understanding the domain and range of functions is crucial in mathematics, especially when dealing with inverse functions. The **domain** of a function is the set of all possible inputs (or 'x' values). On the other hand, the **range** is the set of all possible outputs (or 'y' values).

For two functions to be inverses, the domain of one function becomes the range of the other and vice versa. This interchanging of domain and range is fundamental when proving two functions are inverses through composition.

• For \( f(x) = \frac{3}{4}x - 2 \), if we list all the 'x' values it can accept, that's its domain. Similarly, the 'y' values that result from these inputs make up the range.
• The function \( g(x) = \frac{4}{3}x + \frac{8}{3} \) will have its domain and range swap roles with \( f \), if they are indeed inverse functions.

Hence, identifying how these domains and ranges interact is paramount when dealing with inverse functions.

Algebraic Functions

Algebraic functions are a broad class of functions built using algebraic expressions. This means they can include operations like addition, subtraction, multiplication, division, and taking roots. The functions \( f(x) \) and \( g(x) \) we used are simple examples of linear algebraic functions, where each function is essentially a straight line when graphed.

These linear functions are characterized by constants and coefficients that determine their slope and y-intercept. For example:

• \( f(x) = \frac{3}{4}x - 2 \) has a slope of \( \frac{3}{4} \) and a y-intercept of \(-2\).
• \( g(x) = \frac{4}{3}x + \frac{8}{3} \) has a slope of \( \frac{4}{3} \) and a y-intercept of \( \frac{8}{3} \).

Linear functions are straightforward and help illustrate the concept of inverse functions effectively, as their inverses, if they exist, can also be expressed in a similar linear form.