Problem 61 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The value of \(g(f(2))\) is 2

Step 1: Find the value of \(f(2)\)

To find \(f(2)\), substitute \(x=2\) into the function \(f\). \[f(x)=\frac{x-2}{3}\] \[f(2)=\frac{2-2}{3}\]

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

\[f(2)=\frac{0}{3}=0\]

Step 3: Substitute \(f(2)\) into \(g(x)\)

Now substitute the result obtained for \(f(2)\) into \(g(x)\).\n\[g(f(2))=g(0)=3*0+2\]

Step 4: Simplify \(g(f(2))\)

After substituting \(x=0\) into \(g(x)\), we get: \[g(0) = 0 + 2=2\]

Key Concepts

Function Evaluation

Function evaluation is the process of finding the output of a function for a given input value. In this particular exercise, we are evaluating functions at specific points. To evaluate a function, you substitute the input value into the function's equation to calculate the result.
For example, to evaluate the function \( f(x) = \frac{x-2}{3} \) at \( x = 2 \), you substitute 2 into the equation, resulting in \( f(2) = \frac{2-2}{3} = \frac{0}{3} = 0 \).
Likewise, with the function \( g(x) = 3x + 2 \), evaluating \( g(0) \) involves substituting 0 in place of \( x \), leading to \( g(0) = 3 \cdot 0 + 2 = 2 \). This process of substitution and simplification is essential for accurately using and interpreting mathematical functions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations. They represent mathematical relationships and can be simplified or evaluated based on given values.
In the exercise, the expressions \( g(x) = 3x + 2 \) and \( f(x) = \frac{x-2}{3} \) are algebraic forms where the letter \( x \) is a variable. To operate with these expressions, you need to follow the order of operations:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

For instance, when evaluating \( f(2) \), we apply the operations within the expression to substitute and solve for \( x = 2 \). By understanding algebraic expressions, students can confidently manipulate and solve them, turning abstract concepts into concrete results.

Function Composition

Function composition involves applying one function to the results of another function. This means creating a new function by combining two existing functions in a specific order. Function composition is denoted by \( (f \circ g)(x) \), meaning you first apply \( g \) to \( x \) and then apply \( f \) to the result of \( g(x) \).
In the exercise, we are tasked with finding \( g(f(2)) \). This involves two main steps:

• First, compute \( f(2) \) using the function \( f(x) = \frac{x-2}{3} \) to find \( f(2) = 0 \).
• Second, take this result (0) and use it as the input for the function \( g \) to find \( g(0) \).

This results in \( g(0) = 2 \). Understanding function composition is crucial as it allows for evaluating complex function relationships efficiently.