Problem 57 - Elementary and Intermediate Algebra

Question

Find \(g(2)\) and \(g(3) .\) See Example 4. $$ g(x)=2 x^{2}-x+1 $$

Step-by-Step Solution

Answer

\( g(2) = 7 \) and \( g(3) = 16 \).

Step 1: Understand the Function

The given function is \( g(x) = 2x^2 - x + 1 \). This is a quadratic function of the form \( ax^2 + bx + c \) where \( a = 2 \), \( b = -1 \), and \( c = 1 \). Our task is to find the values of this function at specific points, namely \( x = 2 \) and \( x = 3 \).

Step 2: Substitute x = 2 into the Function

To find \( g(2) \), substitute \( x = 2 \) into the equation: \[ g(2) = 2(2)^2 - 2 + 1 \].

Step 3: Calculate \( g(2) \)

Calculate the value: \( 2(2)^2 = 2 \times 4 = 8 \). Now substitute into the equation: \[ g(2) = 8 - 2 + 1 \]. Simplify this: \( 8 - 2 = 6 \) and \( 6 + 1 = 7 \). So, \( g(2) = 7 \).

Step 4: Substitute x = 3 into the Function

To find \( g(3) \), substitute \( x = 3 \) into the equation: \[ g(3) = 2(3)^2 - 3 + 1 \].

Step 5: Calculate \( g(3) \)

Calculate the value: \( 2(3)^2 = 2 \times 9 = 18 \). Now substitute into the equation: \[ g(3) = 18 - 3 + 1 \]. Simplify this: \( 18 - 3 = 15 \) and \( 15 + 1 = 16 \). So, \( g(3) = 16 \).

Key Concepts

Function Evaluation

Function evaluation involves finding the output of a function given a specific input. In this context, the function is quadratic, represented as \( g(x) = 2x^2 - x + 1 \). To evaluate this function at particular points (like \( x = 2 \) and \( x = 3 \)), we replace the variable \( x \) in the function expression with the given number.

This process helps us determine the function's specific output, allowing us to understand its behavior at certain inputs. Function evaluation is a foundational concept in algebra and calculus, as it is the basis for analyzing and understanding functions.

Here are some key points to remember about function evaluation:

• Identify the function and the given input value.
• Substitute the input into the function.
• Perform arithmetic operations to simplify the expression.
• Determine the final value of the function at that input.

Substitution Method

The substitution method is a straightforward technique used to replace a variable with a given value in an algebraic expression or function. When solving problems involving functions, like finding \( g(2) \) or \( g(3) \), substitution is key.

In our example, this meant taking the function \( g(x) = 2x^2 - x + 1 \) and substituting \( x = 2 \) in order to determine \( g(2) \), and repeating the process with \( x = 3 \) to find \( g(3) \). Beyond simply replacing variables with values, substitution helps in:

• Converting word problems into solvable equations.
• Reducing quadratic and polynomial expressions for evaluation.
• Simplifying complex expressions methodically.
• Eliminating variables during systems of equations solving.

The beauty of the substitution method lies in its ease of use and wide applicability, not only in simple function evaluations but also in more advanced mathematics topics.

Polynomial Algebra

Polynomial algebra involves working with polynomial expressions, which are mathematical expressions consisting of variables and coefficients. In our example, \( g(x) = 2x^2 - x + 1 \) is a quadratic polynomial, a special type of polynomial with the highest degree of 2.

Key components of polynomial algebra include:

• Identifying terms, coefficients, and degrees of the polynomial.
• Performing algebraic operations like addition, subtraction, and multiplication of polynomials.
• Factoring polynomials for simplification or solving equations.
• Evaluating polynomials at given values, like what was done with \( g(2) \) and \( g(3) \).

Understanding these core aspects provides a strong foundation for tackling more advanced algebra topics. Polynomial equations, like our example, are crucial in modeling real-world situations and understanding a wide variety of mathematical phenomena.

By mastering polynomial algebra, students can better understand concepts like function behavior, graphing, and calculus, ultimately gaining a deeper mathematical intuition.