Problem 57 - Elementary and Intermediate Algebra

Question

Let \(f(x)=3 x-2\) and \(g(x)=x^{2}+x .\) Find each of the following. $$ (f \circ g)(4) $$

Step-by-Step Solution

Answer

(f \circ g)(4) = 58.

Step 1: Understand Composition Functions

The composition of functions \((f \circ g)(x)\) means you apply the function \(g\) first, then use the result as the input for the function \(f\). This can be written as \(f(g(x))\).

Step 2: Evaluate \(g(4)\)

Begin by substituting \(x = 4\) into the given function \(g(x) = x^2 + x\). So, \(g(4) = 4^2 + 4 = 16 + 4 = 20\).

Step 3: Evaluate \(f(g(4))\) or \(f(20)\)

Now use the value found in Step 2 as input for the function \(f(x) = 3x - 2\). Substitute \(x = 20\) into \(f\), so \(f(20) = 3(20) - 2 = 60 - 2 = 58\).

Key Concepts

Understanding Algebraic Expressions in Function Composition

When working with functions in algebra, you often need to manipulate and interpret algebraic expressions. Algebra serves as a foundational language in which we express relationships between variables through equations and formulas. For instance, in composing functions like \( (f \circ g)(x) \), you're essentially performing operations that combine two algebraic expressions.
Algebraic operations include:

• Addition and subtraction - used to combine or separate terms within expressions. Example: in \( g(x) = x^2 + x \), the terms \( x^2 \) and \( x \) are added.
• Multiplication - often used when evaluating expressions within another function, such as multiplying terms when you substitute values. Example: in \( f(x) = 3x - 2 \), you multiply \( x \) by \( 3 \).
• Substitution - involves replacing variables with numbers or other variables to find new expressions or values.

Working through algebra strengthens your ability to solve problems like function compositions by deciphering and simplifying expressions, ultimately leading to an understanding of dynamic mathematical relationships.

Substitution in Function Evaluation

Substitution is a key technique in evaluating function compositions. It involves replacing variables in an expression with specific values or other expressions. By doing so, you transform an abstract formula into a concrete calculation more easily understood.
Here's how substitution works in the given problem:

• Start with the inner function first. For \( g(x) = x^2 + x \), substitute \( x=4 \) to find \( g(4) \):
  • Calculate \( 4^2 + 4 = 16 + 4 = 20 \). This gives you the output of \( g(4) \).
• Next, use this result as the input for the outer function \( f(x) = 3x - 2 \). Substitute \( 20 \) into \( f(x) \):
  • Calculate \( f(20) = 3 \times 20 - 2 = 60 - 2 = 58 \).

Substitution simplifies the task by breaking it into manageable steps and is essential when working with nested functions, as it systematically reduces complexity.

The Process of Function Evaluation

Evaluating functions, particularly compositions like \( (f \circ g)(x) \), is about processing input through multiple layers: each function acts upon the results of the previous one. Let's understand the general steps involved in function evaluation:
In our exercise, to find \( (f \circ g)(4) \):

• Identify each function: We have \( f(x) = 3x - 2 \) and \( g(x) = x^2 + x \).
• Work inside out: Start with \( g(x) \), since it will transform your initial input \( x \). Evaluate \( g(4) \) to get \( 20 \).
• Carry forward the result: Take the outcome from \( g(x) \) and use it as the new input for \( f(x) \). Substitute \( 20 \) into \( f \) to find \( f(20) \).
• Conclude with the outer function: As you complete these steps, you evaluate \( f(20) \), which ultimately gives you the final result, \( 58 \).

Function evaluation is systematic and requires an understanding of order and process—first solving for the inner function and using its output as the input for the next function. This approach can demystify what initially seems complicated.