Problem 7 - 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 \(I\) \(f(1)\)

Step-by-Step Solution

Answer

The value of f(1) is 0

Step 1: Substitute 1 into f(x)

To find the value of the function at a specific point, substitute the given value into the function. Here, substitute 1 into the function f(x). Thus, calculate \[ f(1) = (1)^4 - 1 \]

Step 2: Simplify the Expression

Simplify the expression by performing the calculations. \[ f(1) = 1 - 1 = 0 \] Therefore, \[ f(1) = 0 \]

Step 3: Method 2 - Use Properties of Functions

For polynomial functions like this, sometimes using properties can simplify the calculation. Here, it's straightforward since we have \[ f(1) = 1^4 - 1 \]. This directly calculates as 4^4 = 256. Subtract 1: 256-1= 255 so it was simplified directly

Key Concepts

substitution method

The substitution method is essential when evaluating functions at specific points. In the substitution method, we replace the variable in the function with the given number. This allows us to calculate the function's value at that point.

For example, if we need to find the value of \( f(1) \) for the function \( f(x) = x^4 - 1 \):

• First, replace \( x \) with 1 in the function. This means we rewrite the function as \( f(1) = 1^4 - 1 \)
• Perform the calculation: \( 1^4 = 1 \)
• Subtract 1 from 1: \( 1 - 1 = 0 \)
• Hence, \( f(1) = 0 \)

Using substitution, we directly see how the function behaves at certain points.

This method is straightforward and very useful for different types of functions.

polynomial functions

Understanding polynomial functions is crucial for various mathematical problems. A polynomial function is expressed as a sum of terms where each term includes a variable raised to a whole number power and multiplied by a coefficient.

In our example, \( f(x) = x^4 - 1 \) is a polynomial function. Here are some characteristics of polynomial functions:

• The degree of the polynomial is the highest power of the variable (in our example, 4).
• The terms follow the general form \( ax^n \) where \( a \) is a coefficient and \( n \) is a non-negative integer.
• Polynomial functions are continuous and smooth graphs, making them predictable and manageable.

Polynomial functions can model various realistic scenarios, from simple to highly complex systems. It’s essential to understand how to manipulate and evaluate them for success in calculus and other advanced topics.

simplification

Simplification is the process of refining a mathematical expression to make it easier to handle or understand. In the context of evaluating functions, simplification means performing arithmetic operations step by step until you get a simple and clear result.

For our exercise, after substituting 1 into the function \( f(x) = x^4 - 1 \), we simplify it through calculation:

• First, calculate \( 1^4 \), which is 1.
• Then, subtract 1 from 1 to get 0.

Hence, \( f(1) = 0 \). Simplifying makes it easier to see the result and follow the process.
Simplification isn't limited to arithmetic. It includes combining like terms, factoring where possible, and reducing fractions.

Always aim for the simplest form of an expression to make your calculations efficient and reduce the chance of errors.