Problem 55 - Elementary Algebra

Question

Specify the domain of the equation \(y=\frac{3}{7+x}\).

Step-by-Step Solution

Answer

Answer: The domain of the function is \(x \neq -7\).

Step 1: Identify the denominator of the function

The denominator of the function is the part of the fraction that appears under the line, which in our case is \((7+x)\).

Step 2: Find values of x that make the denominator zero

To find the values of x that would make the function undefined, we need to find the values that would cause the denominator to be equal to zero: \((7+x) = 0\).

Step 3: Solve the equation for x

To solve the equation, subtract 7 from both sides: \(x=-7\).

Step 4: Specify the domain

Now that we know the value of x that causes the denominator to be zero, we can state the domain of the function by specifying all x values except the one that would make the denominator zero: The domain of the function is \(\boxed{x \neq -7}\).

Key Concepts

Algebraic Functions

Understanding the nature of algebraic functions is like getting to know the building blocks of algebra. An algebraic function is a type of function that involves some combination of addition, subtraction, multiplication, division, or exponents where the variables are always powers of whole numbers, such as in the equation y = \frac{3}{7 + x}. These functions can take various shapes like lines, parabolas, or rational curves, and they are fundamental in the study of mathematics. The function provided as an example is a rational function because it represents a ratio of two polynomials. In this case, it's a simple ratio where the numerator is the constant 3, and the denominator is the binomial \(7 + x\).

The domain of algebraic functions consists of all the possible values of \(x\) that will output a real number for \(y\) when the function rule is applied. It is crucial to recognize that certain operations have restrictions; for instance, division by zero is undefined. When we set the denominator equal to zero and solve for \(x\), we find which value(s) to exclude from the domain, ensuring the function remains well-defined.

Denominator of a Fraction

The concept of the denominator of a fraction is relatively simple, yet vitally important. Think of a fraction as a division problem that hasn’t been solved yet. The denominator, which is the number below the fraction bar, represents how many parts the whole is divided into. In the function y = \frac{3}{7 + x}, the denominator is \(7 + x\).

Why does the denominator get so much attention? Because the value of the denominator directly affects the existence and the behavior of the function. In mathematics, division by zero is a major no-no, as it leads to results that are not defined within the realm of real numbers. Consequently, any value that would turn the denominator into zero needs to be excluded from the domain of the function to maintain its integrity. By acknowledging this, it’s clear why we must consider the denominator's values meticulously to avoid undefined situations in algebraic functions.

Undefined Functions

Diving into the realm of undefined functions can be a bit perplexing. An 'undefined function' may sound like an oxymoron, as functions are generally rules that assign each input exactly one output. The twist here is that certain operations can break these rules, leading to an 'undefined' scenario. The classic example is division by zero; it simply doesn't make mathematical sense (it's like asking to share zero cookies with a friend – there’s nothing to share!).

In the context of the exercise y = \frac{3}{7 + x}, we see that the function becomes undefined when \(x = -7\), since this would lead to division by zero. As such, we exclude \(x = -7\) from our domain. This isn’t just about avoiding one bad apple though; it’s about preserving the integrity of the mathematical universe. By understanding and identifying the conditions under which functions become undefined, students can accurately determine the domain of algebraic functions to ensure they always produce meaningful results.