Problem 60 - Algebra 1: Concepts and Skills

Question

Evaluate the function when x 0, 1, 2, 3, and 4. \(f(x)=4 x\)

Step-by-Step Solution

Answer

The values of the function f(x) for x = 0, 1, 2, 3, and 4 are 0, 4, 8, 12, and 16, respectively.

Step 1: Substituting x = 0

Replace x in \(f(x)=4 x\) by 0. This yields \(f(0)=4*0=0\).

Step 2: Substituting x = 1

Replace x in \(f(x)=4 x\) by 1. This gives \(f(1)=4*1=4\).

Step 3: Substituting x = 2

Replace x in \(f(x)=4 x\) by 2. This leads to \(f(2)=4*2=8\).

Step 4: Substituting x = 3

Replace x in \(f(x)=4 x\) by 3. This results in \(f(3)=4*3=12\).

Step 5: Substituting x = 4

Replace x in \(f(x)=4 x\) by 4. This computes to \(f(4)=4*4=16\).

Key Concepts

Function Evaluation

Understanding how to evaluate functions is a critical skill in algebra. It means figuring out what you'll get when you replace the variable in the function with a specific value. To do this, you follow a straightforward process:

First, you identify the function rule, such as in our case, where the function rule is given by the algebraic expression \(f(x) = 4x\). Next, you take the value you want to evaluate—say, \(x = 1\)—and substitute it into the function wherever you see \(x\).

Here, you would calculate \(f(1) = 4 \times 1\) to find that the function evaluates to 4. Using this method for different values of \(x\), such as 0, 2, 3, and 4, results in \(f(0) = 0\), \(f(2) = 8\), \(f(3) = 12\), and \(f(4) = 16\), respectively. Thus, function evaluation is really about substituting and simplifying.

Linear Functions

Linear functions are the simplest type of function you'll encounter in algebra. A linear function is one where the variable x is always to the first power and never appears as a higher power or under a radical or exponent.

The general form of a linear function is \(f(x) = mx + b\), where 'm' represents the slope, and 'b' is the y-intercept of the function's graph on a coordinate plane. In the problem we're tackling, \(f(x) = 4x\) is a linear function because it can be rewritten as \(f(x) = 4x + 0\), showing that 'm' is 4 and 'b' is 0.

This means the graph of this function is a straight line with a slope of 4, which indicates that for each step increase in x, f(x) increases by 4. Evaluating this linear function requires the use of the substitution method as demonstrated previously.

Substitution Method

The substitution method is a fundamental technique in algebra used to find the value of an expression given a specific value of a variable. It involves two main steps. First, you substitute, which means replacing the variable with a given number. Then you simplify the expression by following the order of operations – working out multiplications or divisions before additions or subtractions.

In our problem's context, when you substitute \(x\) with a number (0, 1, 2, etc.), you are literally placing that number in the 'spot' of \(x\) in the function \(f(x) = 4x\). After substitution, you simplify to get the result. It is through this method that one can find the output of functions for given inputs quickly and efficiently.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can have numbers, variables, and operation symbols. These expressions represent values that can vary, which we see when working with functions like \(f(x) = 4x\). They are the backbone of many algebraic concepts, allowing us to describe rules and relationships between quantities that can change.

In algebraic expressions, the variable - in this case, \(x\) - stands in as a placeholder for the input value. The beauty of algebra is that by manipulating these expressions according to established mathematical laws, we can solve for variables, make predictions, and understand the nature of the relationships depicted by the expressions.