Problem 18 - Algebra 2

Question

Find \(p(4)\) and \(p(-2)\) for each function. \(p(x)=2-x\)

Step-by-Step Solution

Answer

p(4) = -2; p(-2) = 4.

Step 1: Substitution for p(4)

Start by substituting 4 into the function: \[ p(x) = 2 - x \] Replace \( x \) with 4: \[ p(4) = 2 - 4 \]

Step 2: Simplification for p(4)

Simplify the expression from the substitution: \[ p(4) = 2 - 4 = -2 \] Thus, \( p(4) = -2 \).

Step 3: Substitution for p(-2)

Next, substitute -2 into the function: \[ p(x) = 2 - x \] Replace \( x \) with -2: \[ p(-2) = 2 - (-2) \]

Step 4: Simplification for p(-2)

Simplify the expression from the substitution: \[ p(-2) = 2 + 2 = 4 \] Thus, \( p(-2) = 4 \).

Key Concepts

Substitution

In mathematics, substitution is a fundamental technique that is used to replace a variable with a specific value. This process is particularly useful when evaluating functions, as it allows you to determine the output for certain inputs. In the context of the exercise, substitution involves replacing the variable \( x \) in the function \( p(x) = 2 - x \) with a number, such as 4 or -2, to find \( p(4) \) or \( p(-2) \).

• The first step is identifying which number will replace the variable \( x \).
• Next, you substitute this number into the function in place of \( x \).

For instance, to find \( p(4) \), substitute 4 for \( x \): \[p(4) = 2 - 4 \]The same logic applies to finding \( p(-2) \), where \(-2\) is substituted for \( x \): \[p(-2) = 2 - (-2)\] This technique simplifies the evaluation by providing clear steps to follow and makes complex expressions easier to handle.

Simplification

Simplification in mathematics is the process of making an expression easier to understand by combining like terms and performing basic arithmetic operations. This step follows substitution and is crucial for obtaining a final, simplified answer.
After substituting a number for a variable, you simplify by performing operations such as addition or subtraction. It helps in obtaining a cleaner, final result that is easy to comprehend.

• Perform arithmetic operations such as addition and subtraction.
• Combine similar terms if necessary.

To simplify \( p(4) = 2 - 4 \), you calculate:\[p(4) = -2\]And for \( p(-2) = 2 - (-2) \), you get:\[p(-2) = 4\]Simplification is essential because it reduces complex expressions into simpler results, making it clearer and more concise.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers. They are one of the simplest and most widely used types of functions in mathematics. The function in our exercise, \( p(x) = 2 - x \), is a linear polynomial, which is the simplest form of a polynomial function where no variable has an exponent greater than one.
Key characteristics of polynomial functions include:

• They can have constants, variables, and exponents.
• The highest power of the variable determines the degree of the polynomial.

In a polynomial like \( p(x) = 2 - x \):- The degree is 1 because \( x \) is raised to the power of 1.- Such functions graph as straight lines.Polynomial functions are fundamental to algebra and calculus, providing the foundation for more advanced studies. Understanding how to evaluate them through substitution and simplification reinforces basic algebraic manipulation skills and aids in solving various practical and theoretical problems.