Problem 56 - A Graphical Approach to Precalculus with Limits

Question

Find \(f(x)\) at the indicated value of \(x\). $$f(x)=3 x^{2}+2 x-5 ; x=2$$

Step-by-Step Solution

Answer

11

Step 1: Identify the function and given value

The problem provides us with a function \(f(x) = 3x^2 + 2x - 5\) and asks us to find its value when \(x = 2\). This means we'll substitute \(x = 2\) into the function and simplify.

Step 2: Substitute the value into the function

Replace \(x\) with \(2\) in the function: \(f(2) = 3(2)^2 + 2(2) - 5\). This will allow us to evaluate the function at \(x = 2\).

Step 3: Calculate the squared term

Compute \(2^2\) to get \(4\). Then substitute this back into the expression: \(f(2) = 3(4) + 2(2) - 5\).

Step 4: Multiply coefficients with their terms

Multiply the coefficients with their respective terms: calculate \(3 \times 4 = 12\) and \(2 \times 2 = 4\). Substitution gives: \(f(2) = 12 + 4 - 5\).

Step 5: Combine like terms

Add the results from the previous step and subtract: \(12 + 4 = 16\). Then, calculate \(16 - 5\) to find \(f(2) = 11\).

Step 6: Conclude the solution

We have simplified the expression to \(f(2) = 11\). Thus, the value of \(f(x)\) when \(x = 2\) is \(11\).

Key Concepts

Substitution

Substitution is a fundamental concept in mathematics that involves replacing a variable in an expression or an equation with a given number or another expression. This is a crucial step in evaluating functions, where you determine the outcome of a function for a specific input value.

In this exercise, we start by identifying the function: \(f(x) = 3x^2 + 2x - 5\). Our goal is to find out what this function equals when \(x = 2\). Here's what happens:

• First, replace every occurrence of \(x\) in the function with \(2\).
• Write it out as \(f(2) = 3(2)^2 + 2(2) - 5\).
• This substitution allows us to transition from a general expression to a specific value.

Substitution simplifies the process of calculating the function's value without solving the entire algebraic structure each time.

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole number exponents and coefficients. They form a significant part of algebraic studies. In this exercise, the polynomial function given is \(f(x) = 3x^2 + 2x - 5\).

• The leading term is \(3x^2\), which indicates a quadratic (degree 2) polynomial.
• The function is composed of three terms: a squared term, a linear term, and a constant.
• Polynomial functions can model various real-world scenarios due to their flexibility in shape and behavior.

Understanding how to evaluate these functions will allow you to predict their values for different inputs, crucial in both academic and real-life problem solving.

When dealing with polynomial functions, it is essential to perform operations in the correct order. This involves following specific arithmetic operations, as we shall see in the next section.

Arithmetic Operations

Arithmetic operations form the basic toolkit for working with polynomial expressions like \(3x^2 + 2x - 5\). Here’s how these operations unfold during function evaluation:

• Squaring a term: In \((2)^2\), you square the variable which results in \(4\).
• Multiplication: Essential after squaring, multiply coefficients by the results. Here, \(3 \times 4 = 12\) and \(2 \times 2 = 4\).
• Addition and Subtraction: Finally, combine these values: add \(12\) and \(4\) and then subtract \(5\) to reach \(11\).

Sticking to the correct order—performing operations inside parentheses first, following arithmetic rules, and combining like terms—is necessary for accurate calculation.

Mastering these basic but vital calculations will streamline your ability to handle complex algebraic problems in a step-by-step methodical manner.