Problem 55 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

\(f(1) = 4\).

Step 1: Understand the Function

We are given the quadratic function \(f(x) = 2x^2 - x + 3\). In this function, \(x\) is the variable, and we want to evaluate this function when \(x = 1\).

Step 2: Substitute the Value of x

To find \(f(x)\) at \(x = 1\), replace \(x\) in the function with 1. This gives the expression \(f(1) = 2(1)^2 - 1 + 3\).

Step 3: Calculate Each Term Separately

First, calculate \(2(1)^2\). Since \((1)^2 = 1\), we have \(2 \times 1 = 2\). Second, \(-x = -1\) because \(x = 1\). Third, the \(+3\) remains unchanged.

Step 4: Combine the Results

Combine the calculated values from each term: \(2 - 1 + 3\). Start with the subtraction: \(2 - 1 = 1\). Then add 3: \(1 + 3 = 4\).

Step 5: State the Final Result

The evaluated value of the function at \(x = 1\) is \(f(1) = 4\).

Key Concepts

Polynomial Substitution

Polynomial substitution is a method used to evaluate functions by replacing the variable in the polynomial expression with a specific value. In the context of a quadratic function like \(f(x) = 2x^2 - x + 3\), we substitute \(x = 1\) to find \(f(1)\). Here's how substitution simplifies the process:

• Identify the function and the value for substitution. In our example, this is \(f(x) = 2x^2 - x + 3\) at \(x = 1\).
• Replace the variable \(x\) in the polynomial with the given number. This changes our equation to \(f(1) = 2(1)^2 - 1 + 3\).

By substituting, you transform the general expression into a specific problem that can be solved via arithmetic.

Step-by-Step Calculation

Breaking down calculations into clear steps ensures accuracy. When tackling an expression like \(f(1) = 2(1)^2 - 1 + 3\), consider following these straightforward steps:

• Calculate Exponents First: Find \((1)^2 = 1\).
• Multiply: Take the result from the exponent and multiply by 2, giving \(2 \times 1 = 2\).
• Subtract: Next, handle the subtraction by computing \(-1\), resulting from \(-x\).
• Add: Finally, perform addition with the constant term \(+3\).

Combine these calculated results to simplify to a final answer, methodically and without errors by computing \(2 - 1 + 3\) step-by-step.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operators that represent a particular value or a relationship between values. In the quadratic function \(f(x) = 2x^2 - x + 3\), each part of the expression plays a role:

• Terms: These include \(2x^2\), \(-x\), and \(+3\), each representing different components of the expression.
• Coefficients: The number in front of \(x\) terms, such as 2 in \(2x^2\), indicates how much that term is scaled by.
• Operations: Plus and minus signs connect the terms, detailing how each component interacts within the expression.

Understanding how to navigate through these expressions is crucial for evaluating functions correctly, as each term and operation impacts the final result, as demonstrated in our substitution process.