Problem 9 - Algebra for College Students

Question

Let \(f(x)=x^{4}-1, g(x)=x^{3}-3 x^{2}+5,\) and \(h(x)=4 x^{4}-\) \(3 x^{2}+3 x-1 .\) Find the following function values by using two different methods. See Example \(I\) $$ f(-2) $$

Step-by-Step Solution

Answer

f(-2) = 15

Step 1: Understand the function definition

The given function is: \[f(x) = x^4 - 1\]

Step 2: Substitute the value into the function

Substitute \(x = -2\) into the function: \[f(-2) = (-2)^4 - 1\]

Step 3: Calculate the exponent

Calculate \[(-2)^4 = 16\]

Step 4: Perform the subtraction

Subtract 1 from 16: \[16 - 1 = 15\]

Key Concepts

Understanding Polynomial Functions

Polynomial functions are made up of terms that include variables raised to whole number powers and their corresponding coefficients.
An example of a polynomial function is \(f(x) = x^4 - 1\).
These functions can have one or more terms.
Specific properties and behaviors of polynomial functions depend on their degree (the highest power of the variable).
The function \(f(x) = x^4 - 1\) is a fourth-degree polynomial, meaning its highest power of the variable \(x\) is 4. As degrees increase, the complexity and possible shapes of their graphs also increase.

The Substitution Method in Function Evaluation

The substitution method involves replacing the variable in the function with a specific value, and then performing the necessary arithmetic operations to find the function's value for that particular input.
Here is a structured way of using the substitution method:
1. **Understand the function definition**: Clearly identify the given function. In this case, \(f(x) = x^4 - 1\).
2. **Substitute the value**: Replace the variable \(x\) in the function with the given specific value. For example, to find \(f(-2)\), substitute \(x = -2\): \(f(-2) = (-2)^4 - 1\).
3. **Perform the calculation**: Follow through with the required arithmetic operations:
\((-2)^4 = 16\)
\(16 - 1 = 15\).
This method helps ensure precision and clarity in calculating function values. Always follow each step diligently to avoid errors.

Basic Algebraic Operations in Function Evaluation

Algebraic operations such as addition, subtraction, multiplication, and exponentiation are crucial for evaluating functions.
Let's break down the required operations in simple terms:
The function to evaluate is \(f(x) = x^4 - 1\).
When finding \(f(-2)\), you substitute \(-2\) for \(x\):
\(f(-2) = (-2)^4 - 1\).
Calculating \((-2)^4\) involves multiplying -2 by itself four times:
\((-2) \times (-2) = 4\)
\((4) \times (-2) = -8\)
\((-8) \times (-2) = 16\).
Now, subtract 1 from 16:
\(16 - 1 = 15\).
By breaking down calculations step-by-step and performing each arithmetic operation sequentially, you can accurately find the function's value.