Problem 42 - Algebra 2

Question

Find each value if \(f(x)=x^{2}-3 x+2\) $$ f(2) $$

Step-by-Step Solution

Answer

The value is 0.

Step 1: Identify the Given Function and Substitute the Value

We are given the function \( f(x) = x^2 - 3x + 2 \). We need to find \( f(2) \). To do this, substitute \( x = 2 \) into the function.

Step 2: Calculate Each Term

Substitute \( x = 2 \) into each term of the function: \[ f(2) = (2)^2 - 3(2) + 2 \]Compute each term separately:- \( (2)^2 = 4 \)- \( -3(2) = -6 \)- The constant term is \(+2\).

Step 3: Perform the Arithmetic Operations

Add the results from each term:\[ f(2) = 4 - 6 + 2 \]Evaluate step-by-step:- First perform \( 4 - 6 \) to get \(-2\).- Then, calculate \( -2 + 2 \) resulting in \( 0 \).

Step 4: Write the Final Answer

The value of the function when \( x = 2 \) is \( 0 \). Therefore, \( f(2) = 0 \).

Key Concepts

Function Evaluation

When we talk about function evaluation, we're referring to the process of finding the value of a function given a specific input value for the variable, usually denoted as \( x \). It's like figuring out what the output will be when you plug in a particular number into a machine (the function). In our exercise, we evaluated the function \( f(x) = x^2 - 3x + 2 \) by plugging in \( x = 2 \).

• The first step is to replace each occurrence of \( x \) in the function with the number 2.
• The next step involves calculating each term of the function individually—just simple arithmetic.
• The last step is to combine these results to find the final value of the function.

Evaluating functions is a fundamental skill in mathematics, providing insights into how inputs affect outputs. Remember to carefully substitute and calculate each term to find the correct solution.

Polynomial Functions

Polynomial functions are a specific type of mathematical expressions involving sums of powers of \( x \), each multiplied by a coefficient. A generic polynomial function looks like this: \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \).

In our example, the function \( f(x) = x^2 - 3x + 2 \) is a polynomial of degree 2 because the highest power of \( x \) is 2. Here's why polynomial functions are important:

• They are versatile and can represent a wide range of scenarios, from simple linear relationships to complex curves.
• They have specific characteristics such as degree, leading coefficient, and constant term that influence their shape and position on a graph.

Understanding polynomial functions helps in analyzing patterns and predicting outcomes.

Algebraic Operations

Algebraic operations refer to the basic mathematical procedures used to manipulate expressions, such as addition, subtraction, multiplication, and division. These operations are crucial for evaluating functions like \( f(x) = x^2 - 3x + 2 \).Let's look at the operations we used:

• Substitution: We replaced \( x \) with 2, showing a specific application of algebraic manipulation.
• Exponentiation: Calculating \( 2^2 \) refers to raising a number to a power, a common task in polynomial functions.
• Multiplication and Addition: Computing \( -3(2) + 2 \) involves straightforward multiplication followed by addition or subtraction, combining terms to simplify expressions.

Mastering these operations will make it easier to understand more complex algebraic processes and solve more challenging mathematical problems. Always work step-by-step to ensure accuracy in calculations.