Problem 12 - 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 \(I\) $$ g(-1) $$

Step-by-Step Solution

Answer

g(-1) = 1

Step 1: Understand the Problem

The objective is to find the value of the function \( g(x) \) when \( x = -1 \). Given the function \( g(x) = x^3 - 3x^2 + 5 \), this means substituting \( x = -1 \) into the function.

Step 2: Substitute \( x = -1 \) into \( g(x) \)

Replace \( x \) with \( -1 \) in the function \( g(x) \): \[ g(-1) = (-1)^3 - 3(-1)^2 + 5 \]

Step 3: Calculate \( (-1)^3 \)

Since \( (-1)^3 = -1 \), replace \( (-1)^3 \) with \( -1 \): \[ g(-1) = -1 - 3(-1)^2 + 5 \]

Step 4: Calculate \( (-1)^2 \)

Since \( (-1)^2 = 1 \), replace \( (-1)^2 \) with \( 1 \): \[ g(-1) = -1 - 3(1) + 5 \]

Step 5: Simplify the Expression

Simplifying the equation: \[ g(-1) = -1 - 3 + 5 \] Combine the terms: \[ g(-1) = 1 \]

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression consisting of variables and coefficients combined using addition, subtraction, and multiplication. These expressions are noted for their terms taking the form of \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\), where \(a_n, a_{n-1}, ..., a_1, a_0\) are constants, and \(x\) is the variable. The highest power of the variable (i.e., \(n\)) determines the degree of the polynomial.

For example, consider the polynomial function \(g(x) = x^3 - 3x^2 + 5\). Here, the highest exponent of \(x\) is 3, making it a third-degree polynomial. Polynomial functions can vary in complexity, from simple linear equations (degree 1) to more intricate expressions with higher degrees.

Polynomial functions hold various applications in physics, engineering, economics, and other fields due to their versatile nature in modeling real-world phenomena. Understanding how to work with polynomials is foundational to advancing in mathematics.

Substitution Method

The Substitution Method simplifies evaluating functions by substituting a specific value for the variable in the given equation. This approach is especially helpful when working with polynomial functions, as it makes calculating function values straightforward.

To illustrate, let's use the substitution method in evaluating the function \(g(x) = x^3 - 3x^2 + 5\) at \(x = -1\). The steps involved are:

• Substituting \(x = -1\) into the function: \(g(-1) = (-1)^3 - 3(-1)^2 + 5\).
• Performing the necessary calculations:
• \((-1)^3 = -1 \)
• \((-1)^2 = 1\)
• Simplifying: \(-1 - 3(1) + 5 = -1 - 3 + 5 = 1\)

By substituting \(x = -1\) into the function and simplifying the resulting expression, we derive the value of the function at that point, which is \(g(-1) = 1\). This exemplifies the efficiency of the substitution method in function evaluation.

Function Evaluation

Function evaluation involves finding the output value of a mathematical function for a particular input value. In the context of polynomial functions, this means substituting the input value into the polynomial and simplifying the expression.

Let's revisit our example function \(g(x) = x^3 - 3x^2 + 5\). To find \(g(-1)\):

• Replace every instance of \(x\) with \(-1\):
• \(g(-1) = (-1)^3 - 3(-1)^2 + 5\)”
• Simplify the expression step by step:
• \((-1)^3 = -1\)
• \((-1)^2 = 1\)

Collating the simplified components results in:
\(g(-1) = -1 - 3 + 5 = 1\) Therefore, the function's value at \(x = -1\) is \(1\).

Grasping function evaluation is essential across various fields of mathematics and applied sciences, as it's often a primary step in solving more complex problems.