Problem 41 - Algebra 2

Question

Find each value if \(f(x)=x^{2}-3 x+2\) $$ f(0) $$

Step-by-Step Solution

Answer

The value is 2.

Step 1: Understand the Problem

The exercise asks to find the value of the function \(f(x)\) when \(x = 0\). The function given is \(f(x) = x^2 - 3x + 2\). To solve this, we need to substitute \(x = 0\) into the function.

Step 2: Substitute x with 0

Replace \(x\) with 0 in the function formula. The expression becomes: \[ f(0) = 0^2 - 3 \times 0 + 2 \]

Step 3: Simplify the Expression

Calculate the expression \[ f(0) = 0 - 0 + 2 \] which simplifies directly to 2.

Step 4: Write the Final Answer

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

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function that involves powers of the variable up to the second degree. Specifically, it is expressed in the form:

• \( f(x) = ax^2 + bx + c \)

In this equation, \(a\), \(b\), and \(c\) are constants, with \(a\) not equal to zero. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of \(a\).
Quadratic functions are crucial in various fields, including physics and engineering, because they describe many natural phenomena and systems, such as projectile motion. In this exercise, the given function \(f(x) = x^2 - 3x + 2\) is a typical example of a quadratic function.

Substitution Method

The substitution method is a technique often used in algebra to solve equations and evaluate functions. It involves replacing a variable with its given value or expression.
By substituting, we transform the original problem into a simpler one that is easier to solve. In the context of the exercise provided, we were given \( x = 0 \). By substituting zero into the quadratic function \( f(x) = x^2 - 3x + 2 \), we computed \(f(0)\) as follows:

• Replace \(x\) with 0 in the function: \( f(0) = 0^2 - 3\times0 + 2 \).
• Simplify to find the result: \( f(0) = 2 \).

This step-by-step substitution helps in clear understanding and accurate calculations, particularly in more complex functions.

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. These functions can be linear, quadratic, cubic, or of higher degree.
The general form of a polynomial function is:

• \( P(x) = a_nx^n + a_{n-1}x^{n-1} + \, \ldots \, + a_1x + a_0 \)

Here, the coefficients \(a_n, a_{n-1}, ..., a_0\) are constants, and \(n\) is a non-negative integer. Polynomial functions are foundational in algebra and calculus due to their versatility in modeling relationships and solving equations.
The quadratic function \(f(x) = x^2 - 3x + 2\) from the exercise is specifically a second-degree polynomial function, characterized by its highest power, \(x^2\). Each polynomial term influences the shape and attributes of its graph.

Function Notation

Function notation is a way to represent functions in mathematics succinctly. It uses a letter, typically \(f\), followed by parentheses containing the variable.
This notation indicates the rule applied to the variable, specifying that \(f\) is a function of \(x\). For example, \(f(x) = x^2 - 3x + 2\) means we have a rule defined by \(x^2 - 3x + 2\), which acts on \(x\).

• The notation \(f(0)\) specifies that we are evaluating function \(f\) at \(x=0\).
• Function notation is crucial for clarity in communication, enabling mathematicians to describe and work with complex expressions easily.

Understanding this notation is essential when dealing with functions, as it clearly denotes the relationship and operations performed on the input value. It is a fundamental concept taught early in learning functions and algebra.