Problem 63 - Intermediate Algebra

Question

Let \(f(x)=3 x-2\) and \(g(x)=x^{2}+x .\) Find each of the following. $$ (g \circ f)(x) $$

Step-by-Step Solution

Answer

\((g \circ f)(x) = 9x^2 - 9x + 2\).

Step 1: Understand Composition of Functions

You are given two functions: \(f(x) = 3x - 2\) and \(g(x) = x^2 + x\). The notation \((g \circ f)(x)\) represents the composition of these functions, meaning you first apply \(f(x)\) and then apply \(g(x)\) to the result.

Step 2: Compute \(f(x)\)

Start by determining the expression for \(f(x) = 3x - 2\). This expression will be substituted into the next function \(g(x)\).

Step 3: Substitute \(f(x)\) into \(g(x)\)

Take \(f(x) = 3x - 2\) and substitute it into \(g(x)\). Now, \(g(f(x)) = g(3x - 2)\).

Step 4: Evaluate \(g(3x - 2)\)

Substitute \(3x - 2\) into the function \(g(x) = x^2 + x\). This gives us: \[g(3x - 2) = (3x - 2)^2 + (3x - 2)\].

Step 5: Expand \((3x - 2)^2\)

Expand \((3x - 2)^2\) using the formula \((a - b)^2 = a^2 - 2ab + b^2\): \[(3x - 2)^2 = 9x^2 - 12x + 4\].

Step 6: Combine the Terms in \(g(3x - 2)\)

Add the terms together from both parts of \(g(3x - 2)\) now: \[9x^2 - 12x + 4 + 3x - 2 = 9x^2 - 9x + 2\].

Step 7: Write Final Expression

The final result of \((g \circ f)(x)\) is: \[(g \circ f)(x) = 9x^2 - 9x + 2\].

Key Concepts

Composition of Functions

The composition of functions is an important concept in mathematics that helps simplify complex expressions involving multiple functions. When we compose functions, we are essentially nesting one function inside another. Let's say you have two functions: \(f(x)\) and \(g(x)\). The notation \((g \circ f)(x)\) indicates that you first apply the function \(f\) to \(x\), and then, using the result, apply the function \(g\).
For example, if \(f(x) = 3x - 2\) and \(g(x) = x^2 + x\), finding \((g \circ f)(x)\) requires calculating \(f(x)\) first and then substituting this result into \(g(x)\). This step-by-step approach ensures each function is applied correctly and yields accurate results.
Using this process can make otherwise difficult problems more manageable, as you're only ever working with one part of the problem at a time.

Evaluate Functions

Evaluating functions is the process of plugging a specific value or expression into a function to get an output. Let's consider a simple function like \(f(x) = 3x - 2\). Evaluating this function for a particular value of \(x\), say \(x = 4\), involves substituting \(4\) in place of \(x\):

• \(f(4) = 3(4) - 2 = 12 - 2 = 10\)

This principle can also apply to more complex expressions, such as when we deal with function composition. You must first determine the inner function's result before substituting it into the outer function, just like following a recipe step by step.
In the context of our solution, after finding \(f(x) = 3x - 2\), we evaluated the function \(g\) by plugging the entire expression \(3x - 2\) into \(g(x)\) — knowing each step intimately helps ensure the process is correctly followed and that the output is reliable.

Function Substitution

Function substitution is a technique where one function replaces its variable with another complete function. This technique is particularly useful when evaluating composite functions, and it's necessary for complex analysis and many applications in science and engineering.
Take the functions \(f(x) = 3x - 2\) and \(g(x) = x^2 + x\). To find the composition \(g(f(x))\), function substitution means inserting the outcome of \(f(x)\) into every instance of \(x\) in \(g(x)\). This transforms \(g(x)\) into \(g(3x - 2)\). Each \(x\) in \(x^2 + x\) is replaced by \(3x - 2\):

• \((3x - 2)^2 + (3x - 2)\)

Following this careful approach allows for a systematic integration of functions, often simplifying the problem-solving process by breaking it down into manageable parts.