Problem 71 - Algebra 2

Question

Find each function value. \(f(x)=2 x, f(1)\)

Step-by-Step Solution

Answer

The function value \( f(1) \) is 2.

Step 1: Identify the Function Expression

The problem provides us with a linear function, given as \( f(x) = 2x \). This function indicates that \( f(x) \) is obtained by multiplying any input \( x \) by 2.

Step 2: Substitute the Given Input into the Function

We need to find \( f(1) \), meaning we substitute \( x = 1 \) into the function. Hence, \( f(1) = 2 \times 1 \).

Step 3: Perform the Multiplication

Multiply the terms to solve for \( f(1) \). Calculate \( 2 \times 1 = 2 \).

Step 4: Conclude the Function Value

The value of the function when \( x = 1 \) is evaluated, resulting in \( f(1) = 2 \).

Key Concepts

Function Evaluation

Evaluating a function involves finding the output value when you input a specific value into the function. In our example, the function provided is a linear function: \( f(x) = 2x \).
This formula tells us that for any value of \( x \), we need to multiply it by 2 to find \( f(x) \). Function evaluation is about substituting the given input into the function and calculating the output.
When evaluating \( f(1) \), it simply means plugging in the value of 1 for \( x \). We do this by substituting 1 into the equation, resulting in \( f(1) = 2 \times 1 \).
This step is crucial because it allows us to calculate and understand how changes in the input \( x \) affect the function's output.

Substitution Method

The substitution method is a powerful tool in algebra, commonly used when evaluating functions. It involves replacing the variable in a given function with a specific number or value.
In the problem, we replaced \( x \) with 1 in the function \( f(x) = 2x \).
Here's how it works:

• Identify the variable you are substituting (in this case, \( x \)).
• Replace every instance of that variable in the equation with the given number (here, 1).
• Perform the necessary calculations to find the result.

By substituting \( x \) with 1, we are directly determining the function's output for that specific input value, making it an essential process in analyzing algebraic expressions.

Basic Arithmetic

Basic arithmetic operations are fundamental to solving equations and evaluating functions. They include addition, subtraction, multiplication, and division.
In the context of the problem, we used multiplication, which is one of these core operations.
When we evaluate the function \( f(x) = 2x \) at \( x = 1 \), we calculate \( 2 \times 1 \).
This involves multiplying the coefficient 2 by the input value 1 to get the result.
Understanding and practicing basic arithmetic ensures you can accurately perform these calculations, forming the building blocks for more complicated math problems.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations. In this problem, the expression is \( 2x \) within the function \( f(x) = 2x \).
The term \( 2x \) is composed of the coefficient 2 and the variable \( x \), which together signify multiplication.
Working with algebraic expressions requires understanding how to manipulate these components to solve equations or evaluate functions.

• Coefficients are numbers multiplied by variables.
• Variables represent unknown or variable quantities.
• Arithmetic operations define how the expressions combine.

By mastering these elements, you can effectively analyze and work with more complex mathematical constructs, making algebraic expressions a foundational part of algebra and higher-level mathematics.