Problem 7 - College Algebra

Question

Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$ f(x)=\frac{3}{x-4} \text { and } g(x)=\frac{3}{x}+4 $$

Step-by-Step Solution

Answer

After simplification, we find that neither \(f(g(x))\) nor \(g(f(x))\) equals to \(x\). Hence, these two functions are not inverses of each other.

Step 1: Find \(f(g(x))\)

The composition of two functions \(f\) and \(g\) is the function which pairs the input \(x\) with the output \(f(g(x))\). We replace \(g(x)\) in function \(f\) with the given expression for \(g(x)\), which is \(\frac{3}{x}+4\).\n Now, \(f(g(x))= f(\frac{3}{x}+4)= \frac{3}{\left(\frac{3}{x}+4\right)-4}\). Simplifying this expression will give us \(f(g(x))\).

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

By using the same logic, we replace \(f(x)\) in function \(g\) with \(\frac{3}{x-4}\). So, \(g(f(x)) = g\left(\frac{3}{x-4}\right) = \frac{3}{\left(\frac{3}{x-4}\right)}+4\). Simplifying this will yield \(g(f(x))\).

Step 3: Check if \(f\) and \(g\) are inverses

To prove that two functions are inverses, we need to show that \(f(g(x)) = g(f(x)) = x\). If this is not the case, then the functions are not inverses. To test this, we compare our results from Step 1 and Step 2 to \(x\).

Key Concepts

Function Composition

Function composition is like combining two different functions into one. Imagine you have two separate functions, \(f(x)\) and \(g(x)\). When you compose them, you create a new function, \(f(g(x))\), that first applies \(g\) and then \(f\). This means you substitute the entire \(g(x)\) expression into \(f\).
For instance, if \(f(x) = \frac{3}{x-4}\) and \(g(x) = \frac{3}{x} + 4\), to find \(f(g(x))\), you replace every instance of \(x\) in \(f\) with \(g(x)\). Here, it results in \(f(g(x)) = \frac{3}{\left(\frac{3}{x} + 4\right) - 4}\).

• The first function, \(g(x)\), gets solved first.
• Then, you take its output and plug it into the second function, \(f(x)\).

This concept helps to predict system behaviors in various situations, like tracking changes or conversions.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. Understanding them is essential when working with functions like \(f\) and \(g\).
In our exercises, you deal with expressions such as \(\frac{3}{x-4}\) and \(\frac{3}{x}+4\). By replacing \(x\) in these, you create new expressions like in function compositions.

• Variables can vary, and they usually represent unknown values.
• Operations like addition, subtraction, multiplication, and division are used.

Manipulating expressions, such as rearranging or simplifying, is key to solving equations.
These skills allow you to handle and solve complex problems more effectively.

Simplifying Functions

Simplifying functions means making them easier to work with. It involves reducing expressions to their simplest form.
For our example with \(f(g(x)) = \frac{3}{\left(\frac{3}{x} + 4\right) - 4}\), you simplify by performing basic algebra to reduce the expression. Maybe combine like terms or cancel out factors.

• Calculate and simplify within parentheses first.
• Identify and cancel out common factors if possible.

These simplified forms help to compare, solve, and analyze functions efficiently. If you can simplify until you have just \(x\), it may indicate that you're looking at inverse functions.
This process isn't just for math class—it's useful in everyday scenarios, like simplifying calculations to save time.