Problem 64 - Algebra 2

Question

For each pair of functions, find \(f(g(x))\) and \(g(f(x))\) $$ f(x)=x+3, g(x)=x-5 $$

Step-by-Step Solution

Answer

The function composition \(f(g(x))\) equals \(x-2\) and the function composition \(g(f(x))\) also equals \(x-2\).

Step 1: Find the function composition \(f(g(x))\)

First, substitute \(g(x)\) into \(f(x)\). This involves replacing every instance of \(x\) in \(f(x)\) with the function \(g(x)\), which is \(f(g(x)) = (x-5)+3\.

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

Solve the equation \(f(g(x)) = (x-5)+3\) to simplify it. This gives us \(f(g(x)) = x-2\).

Step 3: Find the function composition \(g(f(x))\)

Then, substitute \(f(x)\) into \(g(x)\). It involves replacing every instance of \(x\) in \(g(x)\) with the function \(f(x)\), resulting in \(g(f(x)) = (x+3)-5\) .

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

Solve the equation \(g(f(x)) = (x+3)-5\) to simplify it. This gives us \(g(f(x)) = x-2\).

Key Concepts

Substitution

Substitution is the process of replacing a variable with another expression. It is a key concept during function composition, where we replace every instance of a variable in one function with another function.
By substituting, we effectively "plug in" the second function into the first. In this exercise:

• For the function composition \( f(g(x)) \), we replace \( x \) in \( f(x) = x + 3 \) with \( g(x) = x - 5 \).
• For the function composition \( g(f(x)) \), we do the reverse by replacing \( x \) in \( g(x) = x - 5 \) with \( f(x) = x + 3 \).

This step is critical for understanding how functions interact with each other through direct replacement. The result is a new function that expresses a relationship derived by combining the two original functions.

Function Operations

Function operations include substitution, addition, subtraction, multiplication, and division of functions. The focus in this exercise is on composition, a form of operation where one function is applied to the result of another.
Here's how it works for our example:

• Function \( f(x) = x + 3 \) is applied to the output of function \( g(x) = x - 5 \), creating a new expression \( f(g(x)) \).
• Similarly, function \( g(x) \) is applied to the output of function \( f(x) \), leading to \( g(f(x)) \).

The composition operation rearranges how the values change through the application of two functions in succession. Function operations like these are fundamental because they enable us to build more complex functions from simpler ones, enhancing our ability to analyze various mathematical and real-world applications.

Algebraic Simplification

Algebraic simplification involves rewriting mathematical expressions in a simpler or more efficient form. This is a crucial step after function composition to ensure the expressions remain manageable and comprehensible.
In the exercise:

• We found \( f(g(x)) = (x-5) + 3 \). Simplifying this by combining like terms results in \( f(g(x)) = x - 2 \).
• Similarly, for \( g(f(x)) \), the expression \( (x+3) - 5 \) simplifies to \( g(f(x)) = x - 2 \).

The reduction and simplification of expressions help eliminate unnecessary terms and reveal the simplest equivalence. This is important as it offers clearer insights into the behavior and properties of the combined functions. Simplification is not only useful for neatness but also for further calculations and interpretations in problem-solving contexts.