Problem 6 - 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. $$ f(x)=3 x-7 \text { and } g(x)=\frac{x+3}{7} $$

Step-by-Step Solution

Answer

The composite functions are \(f(g(x)) = \frac{3x - 40}{7}\) and \(g(f(x)) = \frac{3x - 4}{7}\). These are not equal to x, hence f and g are not inverses of each other.

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

Start by replacing all instances of \(x\) in the function \(f(x) = 3x - 7\) with \(g(x) = \frac{x + 3}{7}\).\n\nThis gives us \(f(g(x)) = 3\left(\frac{x + 3}{7}\right) - 7\). Simplify this expression to find \(f(g(x))\).

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

When simplifying the expression we get \(f(g(x)) = \frac{3(x + 3)}{7} - 7\). This simplifies further to \(f(g(x)) = \frac{3x + 9}{7} - 7 = \frac{3x + 9 - 49}{7} = \frac{3x - 40}{7}\).

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

Now, let's find \(g(f(x))\). Start by replacing all instances of \(x\) in the function \(g(x) = \frac{x + 3}{7}\) with \(f(x) = 3x - 7\).\n\nThis gives us \(g(f(x)) = \frac{3x - 7 + 3}{7}\). Now, simplify this expression to find \(g(f(x))\).

Step 4: Simplify \(g(f(x))\)

When we simplify the expression we get \(g(f(x)) = \frac{3x - 4}{7}.\n\n

Step 5: Comparison

Two functions \(f\) and \(g\) are inverses of each other if \(f(g(x)) = x\) and \(g(f(x)) = x\) for all \(x\). From the above step, we have \(f(g(x)) = \frac{3x - 40}{7}\) and \(g(f(x)) = \frac{3x - 4}{7}\), both of which are not equal to \(x\). Therefore, the functions \(f\) and \(g\) are not inverses of one another.\n\n

Key Concepts

Composition of Functions

When approaching problems involving the composition of functions, it is important to break down what is meant by composing functions in the first place. This concept essentially revolves around plugging one function into another. You have two functions, say, \(f(x)\) and \(g(x)\), and you combine them to create a new function, \(f(g(x))\).

• The innermost function, \(g(x)\), is calculated first.
• Its result is then plugged into the outer function, \(f(x)\).

In our exercise, to find \(f(g(x))\), we insert \(g(x) = \frac{x + 3}{7}\) into \(f(x) = 3x - 7\). This process might seem tricky at first, but remember that you're just replacing \(x\) in \(f(x)\) with the entire \(g(x)\) function.
Similarly, for \(g(f(x))\), you substitute \(f(x) = 3x - 7\) into \(g(x) = \frac{x + 3}{7}\). Keep these steps clear in your mind: replace, simplify, and analyze the result.

Algebraic Expressions

Algebraic expressions are a fundamental part of working with functions, especially when you need to apply operations like substitution during function composition.An algebraic expression may include numbers, variables, and arithmetic operations such as addition or multiplication.

• For example, \(f(x) = 3x - 7\) is a linear algebraic expression.
• Likewise, \(g(x) = \frac{x + 3}{7}\) is another expression involving division.

Handling these expressions involves substituting variables, performing basic arithmetic operations, and rearranging them to simplify or find solutions. In our composition task, algebraic expressions were substituted and simplified to find both \(f(g(x))\) and \(g(f(x))\). By recognizing and manipulating these expressions correctly, we determine critical aspects of functions, such as whether two functions are inverses.

Simplifying Expressions

After substituting one function into another, the next crucial step involves simplifying the algebraic expression obtained. Simplification makes the expression easier to work with and helps reveal deeper insights into its behavior.To illustrate, let’s simplify an expression like \(f(g(x)) = 3\left(\frac{x + 3}{7}\right) - 7\). By taking one small component at a time:

• First, distribute the \(3\) through \((\frac{x + 3}{7})\).
• Next, combine the results: \(3(x + 3)\) becomes \(3x + 9\).
• Finally, continue simplifying by completing the subtraction with \(7\).

In the end, \(f(g(x))\) simplifies to \(\frac{3x - 40}{7}\). Similarly, for \(g(f(x))\), the process leads to \(\frac{3x - 4}{7}\).Each step plays a vital role in potentially identifying key relationships, such as determining if functions are inverse to each other based on whether their compositions resolve to the identity function, \(x\).