Problem 11 - Algebra 2 and Trigonometry

Question

In \(8-13\) , the domain of \(f(x)=4-2 x\) and of \(g(x)=x^{2}\) is the set of real numbers and the domain of \(h(x)=\frac{1}{x}\) is the set of non-zero real numbers. a. Write each function value in terms of \(x\) . b. Find the domain of each function. \(\left(\frac{g}{f}\right)(x)\)

Step-by-Step Solution

Answer

The function \(\left(\frac{g}{f}\right)(x)\) is \(\frac{x^2}{4 - 2x}\) with domain \((-\infty, 2) \cup (2, \infty)\).

Step 1: Express Each Function in Terms of x

The problem already provides each function in terms of \(x\).- For \(f(x)\), it is given by \(f(x) = 4 - 2x\).- For \(g(x)\), it is provided as \(g(x) = x^2\).- For \(h(x)\), it is \(h(x) = \frac{1}{x}\).

Step 2: Define the Function \(\left(\frac{g}{f}\right)(x)\)

The function \(\left(\frac{g}{f}\right)(x)\) is defined as the quotient of \(g(x)\) divided by \(f(x)\):\[\left(\frac{g}{f}\right)(x) = \frac{g(x)}{f(x)} = \frac{x^2}{4 - 2x}\].

Step 3: Identify the Domain of \(g(x)\)

The domain of \(g(x) = x^2\) is all real numbers because a square is defined for any real number. Therefore, the domain of \(g(x)\) is \((-\infty, \infty)\).

Step 4: Identify the Domain of \(f(x)\)

The domain of \(f(x) = 4 - 2x\) is all real numbers, as linear functions are defined for every real number. Therefore, the domain of \(f(x)\) is \((-\infty, \infty)\).

Step 5: Determine the Domain of \(\left(\frac{g}{f}\right)(x)\)

For the function \(\left(\frac{g}{f}\right)(x)\), the expression \(4 - 2x\) must not be zero, as division by zero is undefined. Solving for \(4 - 2x = 0:\)\[4 = 2x \2 = x\]Therefore, \(x eq 2\). The domain of \(\left(\frac{g}{f}\right)(x)\) is therefore all real numbers except \(x = 2\), represented as \((-\infty, 2) \cup (2, \infty)\).

Key Concepts

Rational Functions

Rational functions are fascinating mathematical expressions that consist of two polynomials, one in the numerator and the other in the denominator. In other words, a rational function looks like a fraction where both the top and bottom are polynomials. The general form of a rational function can be written as \( R(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials.

An important aspect of rational functions is their domain, which is determined by the denominator. Since division by zero is undefined, any values of \( x \) that make the denominator zero are excluded from the domain. Consequently, to find the domain of a rational function, you simply set the denominator equal to zero and solve for \( x \). These solutions indicate the values that \( x \) cannot take.

For example, in the function \( \left(\frac{g}{f}\right)(x) = \frac{x^2}{4 - 2x} \), the denominator \( 4 - 2x \) cannot be zero. Solving \( 4 - 2x = 0 \) gives \( x = 2 \). Thus, the domain is all real numbers except \( x = 2 \), represented as \( (-\infty, 2) \cup (2, \infty) \).

Real Numbers

The set of real numbers is one of the most foundational concepts in mathematics. Real numbers include all the numbers that can be found on the number line. They encompass a wide variety of numbers such as:

• Rational numbers, which are numbers that can be expressed as fractions like \( \frac{3}{4} \) or \(-2\).
• Irrational numbers, such as \( \pi \) and \( \sqrt{2} \), which cannot be expressed as simple fractions and have non-repeating, non-terminating decimal expansions.
• Whole numbers and integers, like \( 0, 1, -17 \), which cover natural numbers and their negative counterparts not including fractional parts.

These numbers form a continuous line, meaning they fill all the gaps between integers and include every possible decimal number you can imagine. The real numbers are denoted by the symbol \( \mathbb{R} \), and the domain of many functions, such as \( f(x) = 4 - 2x \) and \( g(x) = x^2 \), is all real numbers, unless specified otherwise.

Understanding real numbers is crucial especially when determining the domains of functions, as it allows us to identify limitations and exclusions from all possible real values.

Division by Zero

Division by zero is a topic that often raises questions because it involves a fundamental rule in mathematics that we cannot divide anything by zero. When dealing with rational functions or any expression where there is a division, ensuring the denominator is never zero is crucial.

Why can't we divide by zero? Imagine dividing an apple pie into zero pieces. It's nonsensical because you can't divide something into pieces that don't exist. Mathematically, when we attempt to divide a number by zero, it leads to undefined behavior. That's why divisions like \( \frac{1}{0} \) are said to be undefined.

In the context of functions, if the denominator becomes zero for any value of \( x \), then that value must be excluded from the function's domain. For example, in \( \left(\frac{g}{f}\right)(x) = \frac{x^2}{4 - 2x} \), the denominator \( 4 - 2x \) should not be zero. Solving \( 4 - 2x = 0 \) gives \( x = 2 \). So, \( x = 2 \) is excluded from the domain as division by zero at \( x = 2 \) would make the function undefined.

Being aware of this restriction helps to avoid making errors when working with functions and calculations, ensuring we only ever work within the defined parameters of mathematics.