Problem 41 - Intermediate Algebra

Question

Find each function value if \(f(x)=3 x^{2}-2 .\) See Section 3.2 $$ f(-1) $$

Step-by-Step Solution

Answer

The value is 1.

Step 1: Identify the function

We are given the function \( f(x) = 3x^2 - 2 \). We want to find the value of the function when \( x = -1 \).

Step 2: Substitute the value

Substitute \( x = -1 \) into the function: \( f(-1) = 3(-1)^2 - 2 \).

Step 3: Calculate the square

Calculate \((-1)^2\), which equals \(1\). Substitute back into the expression: \( f(-1) = 3 \times 1 - 2 \).

Step 4: Multiply and simplify

Multiply the terms: \(3 \times 1 = 3\). Then, the expression becomes \( f(-1) = 3 - 2 \).

Step 5: Final calculation

Finally, subtract \(2\) from \(3\): \( f(-1) = 1 \).

Key Concepts

Understanding Polynomial Functions

A polynomial function is an algebraic expression involving variables raised to whole number exponents and coefficients. Unlike other expressions, it can have multiple terms that vary depending on the exponents. The general form of a polynomial is given by:
\[a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\] where \(a_n, a_{n-1}, ..., a_0\) are constants, and \(n\) is a non-negative integer representing the highest degree of the polynomial.

Polynomial functions can be classified based on their degree:

• Linear: degree 1, like \(f(x) = mx + b\).
• Quadratic: degree 2, such as \(f(x) = ax^2 + bx + c\).
• Cubic: degree 3, such as \(f(x) = ax^3 + bx^2 + cx + d\).

In the given exercise, the function \(f(x) = 3x^2 - 2\) is a quadratic function because the highest power of \(x\) is 2. Polynomial functions play a significant role in algebra as they represent a wide variety of mathematical relationships and allow easy computation through manageable operations.

Substitution Method for Function Evaluation

The substitution method is a straightforward approach used to evaluate functions by replacing the variable with a specific value. This technique simplifies the process of calculating the function's value at any point within its domain.

To apply this method, simply substitute the given \(x\)-value into the function's equation. For example, if you have \(f(x) = 3x^2 - 2\) and need to find \(f(-1)\), the substitution is done by replacing every instance of \(x\) with \(-1\). The equation thus becomes:
\[f(-1) = 3(-1)^2 - 2\]Which is then solved step by step till the final result.

This method is highly effective because it breaks down complex calculations into simple arithmetic, making it easier to evaluate even more intricate functions. It is an essential tool in both learning and applying mathematical concepts in various scenarios.

Working with Algebraic Expressions

Algebraic expressions consist of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. Understanding how to manipulate these expressions is fundamental in algebra and beyond.

In algebraic expressions, often you must perform operations in a specific order to solve them correctly:

• Calculate exponents and powers first.
• Follow with multiplication and division from left to right.
• Finally, perform addition and subtraction from left to right.

For the function \(f(-1) = 3(-1)^2 - 2\) from the original exercise, this involves:

• Squaring \(-1\) to get 1.
• Multiplying 3 by the squared result.
• Subtracting 2 to get the final result.

Mastering these steps ensures accuracy in evaluating algebraic expressions, which is key to solving more complex equations and functioning confidently in algebra and other branches of mathematics.