Problem 56 - Algebra A Combined Function

Question

For each function, find the indicated values. \(g(x)=-\frac{1}{3} x\) a. \(g(0)\) b. \(g(-1)\) C. \(g(3)\)

Step-by-Step Solution

Answer

a. 0, b. 1/3, c. -1

Step 1: Understand the Problem

We are given a linear function \(g(x) = -\frac{1}{3}x\) and need to evaluate this function for specific values of \(x\): 0, -1, and 3.

Step 2: Evaluate g(0)

Substitute \(x = 0\) into the function \(g(x) = -\frac{1}{3}x\). This gives us \(g(0) = -\frac{1}{3} \times 0 = 0\).

Step 3: Evaluate g(-1)

Substitute \(x = -1\) into the function \(g(x) = -\frac{1}{3}x\). This gives us \(g(-1) = -\frac{1}{3} \times (-1) = \frac{1}{3}\).

Step 4: Evaluate g(3)

Substitute \(x = 3\) into the function \(g(x) = -\frac{1}{3}x\). This gives us \(g(3) = -\frac{1}{3} \times 3 = -1\).

Key Concepts

Function Evaluation

Evaluating a function means finding the output for a specific input. In the context of the given exercise, you have a linear function \( g(x) = -\frac{1}{3} x \) and you need to find out its values (outputs) for certain inputs (values of \( x \)). This process is central in mathematics because functions model relationships where each input has precisely one output.

To evaluate a function like \( g(x) \):

• Substitute the given value of \( x \) into the function.
• Perform the arithmetic to find \( g(x) \).

This method helps in determining how changes in \( x \) affect \( g(x) \), providing an understanding of the behavior of the function across its domain.

Substitution Method

The substitution method is an essential skill in algebra. It involves replacing a variable with a given number to simplify an expression or to solve equations. In this exercise, you substitute specific values of \( x \) into the formula \( g(x) = -\frac{1}{3}x \).

Let's see how you apply the substitution method here:

• **For \( g(0) \)**: Replace \( x \) with \( 0 \), then calculate: \( g(0) = -\frac{1}{3} \times 0 = 0 \).
• **For \( g(-1) \)**: Substitute \( -1 \) for \( x \), then solve: \( g(-1) = -\frac{1}{3} \times (-1) = \frac{1}{3} \).
• **For \( g(3) \)**: Insert \( 3 \) into \( x \), hence: \( g(3) = -\frac{1}{3} \times 3 = -1 \).

Substitution provides a systematic way to evaluate functions and solve problems in a step-by-step manner.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operators (like adding or multiplying). They're the building blocks of algebra. An expression such as \( -\frac{1}{3}x \) indicates a linear relationship between \( x \) and \( g(x) \).

When dealing with linear functions:

• **Coefficients**: The number multiplying the variable (e.g., \(-\frac{1}{3}\) in \(-\frac{1}{3}x\)) determines how steep or flat the line is.
• **Variables**: Symbols like \( x \) that stand for numbers.
• **Operations**: Decide how the variable and coefficients interact (e.g., multiplication here).

Understanding algebraic expressions helps in evaluating functions accurately. They reveal what happens within the function as different values of \( x \) are plugged in, making it crucial in analyzing mathematical models.