Problem 15 - 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 \(I\) $$ h\left(\frac{1}{2}\right) $$

Step-by-Step Solution

Answer

0

Step 1: Define the given function

Given the function is \( h(x) = 4x^4 - 3x^2 + 3x - 1 \). The goal is to find \( h \left( \frac{1}{2} \right) \).

Step 2: Substitute \( x = \frac{1}{2} \) into \( h(x) \)

Replace all instances of \( x \) in the function with \( \frac{1}{2} \). This yields: \[ h \left( \frac{1}{2} \right) = 4 \left( \frac{1}{2} \right)^4 - 3 \left( \frac{1}{2} \right)^2 + 3 \left( \frac{1}{2} \right) - 1 \]

Step 3: Simplify the terms inside the function

Calculate each term individually: \( \left( \frac{1}{2} \right)^4 = \frac{1}{16} \) \( \left( \frac{1}{2} \right)^2 = \frac{1}{4} \) Therefore: \[ h \left( \frac{1}{2} \right) = 4 \cdot \frac{1}{16} - 3 \cdot \frac{1}{4} + 3 \cdot \frac{1}{2} - 1 \]

Step 4: Evaluate each term

Now let's evaluate each term: \( 4 \cdot \frac{1}{16} = \frac{4}{16} = \frac{1}{4} \) \( -3 \cdot \frac{1}{4} = -\frac{3}{4} \) \( 3 \cdot \frac{1}{2} = \frac{3}{2} \) Therefore: \[ h \left( \frac{1}{2} \right) = \frac{1}{4} - \frac{3}{4} + \frac{3}{2} - 1 \]

Step 5: Combine like terms

Finally, combine all the evaluated terms: \[ \frac{1}{4} - \frac{3}{4} + \frac{3}{2} - 1 \] Convert all fractions to have a common denominator where necessary and then sum them: \[ \frac{1}{4} - \frac{3}{4} + \frac{6}{4} - \frac{4}{4} = \frac{1 - 3 + 6 - 4}{4} = \frac{0}{4} = 0 \]

Key Concepts

Polynomial Functions

Polynomial functions are expressions that consist of variables, coefficients, and exponents arranged in terms. For example, the given function in the exercise, \( h(x) = 4x^4 - 3x^2 + 3x - 1 \), is a polynomial function of degree 4 because the highest power of the variable \( x \) is 4.
Here are some characteristics of polynomial functions:

• The coefficients (such as 4, -3, 3, and -1) are real numbers.
• The exponents of the variable \( x \) are non-negative integers.
• Polynomial functions are continuous and smooth, meaning they have no breaks or sharp corners in their graphs.

Understanding polynomial functions is crucial for solving and simplifying algebraic equations efficiently.

Substitution Method

The substitution method involves replacing a variable with a given value. In this exercise, we are asked to find \( h\big(\frac{1}{2}\big) \).
This means we will substitute \( x = \frac{1}{2} \) into the function \( h(x) = 4x^4 - 3x^2 + 3x - 1 \) and then simplify.
Here’s a step-by-step overview:

• Identify the function and the given value (in this case, \( x = \frac{1}{2} \)).
• Replace every occurrence of the variable in the function with the given value.
• Simplify the resulting expression to find the value of the function at that specific point.

This method is straightforward and particularly useful for evaluating functions at specific values.

Simplifying Expressions

Simplifying expressions is the process of performing arithmetic operations to reduce an expression to its simplest form. After substituting \( x = \frac{1}{2} \) in the function, the expression \( 4 \big(\frac{1}{2}\big)^4 - 3 \big(\frac{1}{2}\big)^2 + 3 \big(\frac{1}{2}\big) - 1 \) needs to be simplified.
Follow these steps for simplification:

• Calculate each term individually (e.g., \( \big(\frac{1}{2}\big)^4 = \frac{1}{16} \)).
• Apply the coefficients to these calculated values (e.g., \( 4 \times \frac{1}{16} = \frac{1}{4} \)).
• Combine like terms by finding a common denominator, if necessary.
• Simplify the final expression to get the evaluated result (in this case, it simplifies to 0).

Through simplification, we avoid complex fractions and get a clear, final value.

Rational Numbers

Rational numbers are numbers that can be expressed as the ratio of two integers, where the denominator is not zero. Examples include fractions like \( \frac{1}{2} \) or whole numbers like 4 (which can be written as \( \frac{4}{1} \)).
In this exercise, we work extensively with rational numbers while evaluating and simplifying the function. For instance, \( \big(\frac{1}{2}\big)^4 = \frac{1}{16} \) and similarly:\[ -3 \times \frac{1}{4} = -\frac{3}{4} \]

Some key points about rational numbers include:
• They can be positive, negative, or zero.
• They are closed under addition, subtraction, multiplication, and division (except by zero).
• Their decimal representation either terminates or repeats.

Understanding rational numbers is key for performing arithmetic operations involved in function evaluation and simplification.