Problem 59 - Algebra 2

Question

Let \(g(x)=3 x+2\) and \(f(x)=\frac{x-2}{3} .\) Find each value. $$ g(f(-4)) $$

Step-by-Step Solution

Answer

The value of the function composition \(g(f(-4))\) is -4.

Step 1: Find the value of \(f(-4)\)

Substitute \(x = -4\) into \(f(x) = \frac{x - 2}{3}\).\nThus, \(f(-4) = \frac{-4 - 2}{3} = \frac{-6}{3} = -2\).

Step 2: Substitute \(f(-4)\) into \(g(x)\)

Now, substitute \(f(-4)\) which equals -2 into \(g(x) = 3x + 2\).\nSubstituting, we get \(g(f(-4)) = g(-2) = 3(-2) + 2 = -6 + 2 = -4\)

Step 3: Solution

The result of the function composition \(g(f(-4))\) is -4.

Key Concepts

Understanding Functions

In mathematics, a function is a fundamental concept that defines a specific relationship between two sets of elements, typically represented as inputs and outputs. A function assigns exactly one output to each input from its domain.
For example, if we consider the function \( g(x) = 3x + 2 \), \( x \) is the input variable, and \( g(x) \) is the output. Here, the expression \( 3x + 2 \) determines how \( g \) transforms the input \( x \) into an output.

• Functions are often denoted by letters like \( f \), \( g \), and \( h \).
• The input, typically \( x \), belongs to a set called the domain.
• The output, which can be any form of expression, belongs to the codomain.

Functions can be linear, quadratic, or any type where a specific rule applies throughout its domain to produce a consistent result.

Introduction to Linear Functions

Linear functions are a specific type of function where the relationship between the input \( x \) and the output is a straight line when graphed. They have a general form \( f(x) = mx + b \). Here:

• \( m \) is the slope of the line, which indicates the steepness and direction.
• \( b \) is the y-intercept, where the line crosses the y-axis.

For example, in our original problem, the function \( g(x) = 3x + 2 \) is a linear function. The slope \( m \) is 3, signifying that for each increase by 1 unit in \( x \), the value of \( g(x) \) increases by 3. The y-intercept \( b \) is 2, meaning the line intersects the y-axis at the point (0,2).
Linear functions are easy to work with as they provide a constant rate of change second to none in simplicity compared to other functions like quadratic or exponential.

Exploring the Substitution Method

The substitution method is a vital technique in algebra used to solve equations or find the composition of functions. It involves replacing a variable with its known equivalent value, often derived from another equation or function.
In the context of our original exercise, we find \( g(f(-4)) \) by employing the substitution method. Here's how it works:

• First, determine what \( f(-4) \) is by substituting \(-4\) into \( f(x) = \frac{x - 2}{3} \). This gives \( \frac{-6}{3} = -2 \).
• Next, use this result and substitute \(-2\) into \( g(x) = 3x + 2 \), giving \( 3(-2) + 2 = -4 \).

Through substitution, we can find that \( g(f(-4)) = -4 \). The power of this method lies in simplifying many steps into manageable operations, making it easier to understand functions in terms of one another.