Problem 62 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The values of the function at \(x=0\), \(x=1\), \(x=2\), \(x=3\), and \(x=4\) are 1, 4, 7, 10, and 13 respectively.

Step 1: Evaluate the function at x=0

Substitute \(x=0\) into the function: \(f(0)=3(0)+1=1\)

Step 2: Evaluate the function at x=1

Substitute \(x=1\) into the function: \(f(1)=3(1)+1=4\)

Step 3: Evaluate the function at x=2

Substitute \(x=2\) into the function: \(f(2)=3(2)+1=7\)

Step 4: Evaluate the function at x=3

Substitute \(x=3\) into the function: \(f(3)=3(3)+1=10\)

Step 5: Evaluate the function at x=4

Substitute \(x=4\) into the function: \(f(4)=3(4)+1=13\)

Key Concepts

Function Evaluation

To evaluate a function means to find the output or result of a function for a given input. In simpler terms, you input a number into a mathematical expression and find out what it equals. For example, if you have a function like \( f(x) = 3x + 1 \), you can substitute different values for \( x \) to find different results.
When evaluating this function, you replace \( x \) with the number you want to test. For instance, if you want to know the result when \( x = 2 \), you substitute 2 into the function:

• \( f(2) = 3(2) + 1 = 7 \)
• This tells us that the output of the function, or \( f(x) \), is 7 when \( x \) is 2.

This method works for any linear function—a simple way to calculate outputs given an input. Linear functions, like \( f(x) = 3x + 1 \), involve straightforward calculations where each input corresponds to one output.
Understanding function evaluation helps you see how inputs change outputs, giving insight into the function's behavior.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and arithmetic operations (like addition and multiplication). They are the foundation of functions. For instance, \( 3x + 1 \) is an algebraic expression where:

• \( 3x \) means "3 times x," indicating multiplication.
• \( + 1 \) means "add 1," indicating addition.

In a function, the algebraic expression tells us how to change or transform a given input into an output. By understanding how each part works:

• You see that multiplying \( x \) by 3 scales or enlarges the input value.
• Adding 1 shifts the whole result upward by 1 unit.

Operations like these take the input \( x \) and produce an output after performing the specified actions. Mastering algebraic expressions is crucial for all algebra problems because they form the language of math. They let you describe and solve real-world problems systematically. Understanding these expressions lets you evaluate functions more easily, as you become familiar with substituting values and performing operations.

Mathematical Operations

Mathematical operations are the actions you perform within an algebraic expression or equation to solve it or simplify it. The most common operations are addition, subtraction, multiplication, and division. In our function \( f(x) = 3x + 1 \), we see two primary operations:

• **Multiplication:** The term \( 3x \) multiplies the variable \( x \) by 3. This means for every unit change in \( x \), the total changes by 3 units.
• **Addition:** The \( +1 \) adds 1 to the total from the multiplication.

Here's how they work together:

• When you substitute a value into \( x \), start with multiplication to find \( 3x \).
• Then add 1 to whatever you got from \( 3x \).

This order of operations is critical to finding correct results. Always do multiplication before addition in expressions like these. Understanding mathematical operations ensures you can correctly handle and manipulate algebraic expressions and functions. It is the fundamental skill that allows you to evaluate functions by systematically processing each mathematical action.