Problem 55 - A Graphical Approach to College Algebra

Question

Find \(f(x)\) at the indicated value of \(x\). $$f(x)=2 x^{2}-x+3, x=1$$

Step-by-Step Solution

Answer

The value of \( f(1) \) is 4.

Step 1: Understand the Function

The function given is \( f(x) = 2x^2 - x + 3 \). This is a quadratic function, which means it is expressed as \( ax^2 + bx + c \), where in this case \( a = 2 \), \( b = -1 \), and \( c = 3 \). Our task is to find the value of this function when \( x = 1 \).

Step 2: Substitute the Value of x into the Function

We need to find \( f(1) \). Substitute \( x = 1 \) into the function expression: \[ f(1) = 2(1)^2 - 1 + 3 \].

Step 3: Evaluate the Squared Term

Calculate \( 1^2 \): \[ f(1) = 2(1) - 1 + 3 \]. Since \( 1^2 = 1 \), proceed to the next step.

Step 4: Multiply

Multiply \( 2 \) by the result from Step 3: \[ 2 \times 1 = 2 \]. Therefore, the expression becomes \[ f(1) = 2 - 1 + 3 \].

Step 5: Perform Addition and Subtraction

Simplify the expression further:- First, \( 2 - 1 = 1 \).- Then, add \( 1 + 3 = 4 \).So, \( f(1) = 4 \).

Key Concepts

Function Evaluation

In mathematics, evaluating a function means determining the output value when a specific input value is substituted into the function's equation. For example, if given the function \( f(x) = 2x^2 - x + 3 \), to evaluate this function at \( x = 1 \), you find what \( f(1) \) equals.
Function evaluation is a step-by-step process where you:

• Identify the function and its input values.
• Substitute the given input value (i.e., \( x = 1 \)) into the function.
• Follow the order of operations to calculate the function's output value.

This process helps you understand how the function behaves for different input values. You can use this for quadratics, polynomials, or any type of functions.

Substitution Method

The substitution method involves replacing a variable in an equation or expression with a given value. In the quadratic function \( f(x) = 2x^2 - x + 3 \), substituting \( x = 1 \) means you are placing 1 in position of every \( x \) found in the equation.
When substituting, keep these steps in mind:

• Write down the expression or equation clearly.
• Carefully replace the variable \( x \) with the value provided, without altering any other part of the function.
• Follow through with the arithmetic, maintaining the correct order of operations: parentheses, exponents, multiplication/division, and addition/subtraction (PEMDAS).

This technique simplifies the expression, allowing for straightforward calculation of the desired solution.

Polynomial Expressions

Polynomial expressions involve variables raised to different powers summed with coefficients. In this case, \( f(x) = 2x^2 - x + 3 \) is a quadratic polynomial because it features a term with \( x^2 \). Each polynomial has what we call 'terms,' for example:

• The term \( 2x^2 \), where 2 is the coefficient and the power is 2.
• The term \(-x \), which implies a coefficient of -1 and power of 1.
• The constant term +3, which does not have an \( x \) variable.

Understanding polynomials is crucial because they appear frequently in algebra and calculus. They lay the groundwork for solving complex equations and modeling real-world scenarios. Recognizing and manipulating these expressions help in finding solutions efficiently.