Find The Domain Of A Rational Function

Welcome, math enthusiasts! Today, we're diving deep into the concept of the domain of a function, specifically focusing on a rational function. Understanding the domain is crucial because it tells us which input values (x-values) are permissible for a given function. Think of it as the set of "allowed" numbers you can plug into the function without causing any mathematical mishaps, like dividing by zero. Our example function for today is:

$$f(x) = \frac{x+6}{(x-7)(x+5)}$$

To find the domain of any rational function, which is essentially a fraction where the numerator and denominator are polynomials, we need to identify any values of $x$ that would make the denominator equal to zero. Why? Because division by zero is undefined in mathematics, and we want to avoid that at all costs. So, our primary mission is to set the denominator equal to zero and solve for $x$. This will reveal the "forbidden" $x$-values that must be excluded from our domain.

Let's dissect the denominator of our function $f(x)$. It's given by $(x-7)(x+5)$. To find the values of $x$ that make this denominator zero, we set the expression equal to zero:

$$(x-7)(x+5) = 0$$

This equation is a product of two factors. For the entire product to be zero, at least one of the factors must be zero. This leads us to two separate, simpler equations:

1. $x - 7 = 0$
2. $x + 5 = 0$

Solving the first equation, $x - 7 = 0$, we add 7 to both sides to get $x = 7$. This means that if we were to plug $x = 7$ into our function, the denominator would become $(7-7)(7+5) = (0)(12) = 0$. Since we cannot divide by zero, $x = 7$ is not in the domain of $f(x)$.

Now, let's solve the second equation, $x + 5 = 0$. Subtracting 5 from both sides gives us $x = -5$. Similarly, if we plug $x = -5$ into our function, the denominator would be $(-5-7)(-5+5) = (-12)(0) = 0$. Again, division by zero is forbidden, so $x = -5$ is also not in the domain of $f(x)$.

So, we've identified two values, $x = 7$ and $x = -5$, that will cause our function to be undefined. Therefore, the domain of the function $f(x)$ includes all real numbers except for these two values. In interval notation, this would be written as $(-\infty, -5) \cup (-5, 7) \cup (7, \infty)$. In set-builder notation, it's {x \in \mathbb{R} | x \neq -5 and x \neq 7}. This means you can plug in any real number into $f(x)$ as long as it's not -5 or 7.

Let's consider the options provided:

A. All real numbers except 7 B. All real numbers except 5 and -7 C. All real numbers except -5 and 7 D. All real numbers

Based on our calculations, the values that must be excluded are -5 and 7. Therefore, option C accurately describes the domain of the function $f(x)$.

Why is the Domain So Important?

The domain of a function is a fundamental concept in mathematics that dictates the set of all possible input values for which the function is defined and produces a real number output. For rational functions, like the one we just analyzed, the primary restriction on the domain comes from the denominator. Any value of the input variable that causes the denominator to become zero must be excluded. This is because division by zero is an undefined operation. Failing to identify and exclude these values can lead to erroneous calculations and a misunderstanding of the function's behavior.

Consider a scenario where you're modeling a real-world phenomenon with a function. For instance, if you're calculating the time it takes to travel a certain distance at a variable speed, and your function involves a speed in the denominator, you'd instinctively know that a speed of zero is impossible in that context (or would lead to infinite time, which is not a practical answer). The domain helps us understand the practical limitations and valid ranges of our models. In calculus, the concept of domain is essential for understanding continuity, limits, and derivatives. A function can only have a limit or be continuous at a point if that point is within its domain (or at the boundary of an interval in the domain).

Let's explore some other scenarios that can restrict a function's domain:

• Square Roots of Negative Numbers: If a function involves a square root, like $g(x) = \sqrt{x-2}$, the expression inside the square root (the radicand) must be non-negative. In this case, $x-2 \ge 0$, which means $x \ge 2$. So, the domain would be $[2, \infty)$. Any input less than 2 would result in the square root of a negative number, which is not a real number.

• Logarithms: Functions involving logarithms, such as $h(x) = \log(x+3)$, require the argument of the logarithm to be strictly positive. So, $x+3 > 0$, which implies $x > -3$. The domain here is $(-3, \infty)$.

• Trigonometric Functions: Certain trigonometric functions also have restricted domains. For example, the tangent function, $\tan(x) = \frac{\sin(x)}{\cos(x)}$, is undefined whenever $\cos(x) = 0$. This occurs at $x = \frac{\pi}{2} + n\pi$, where $n$ is an integer. Therefore, these values must be excluded from the domain of $\tan(x)$.

Understanding these restrictions allows us to fully characterize a function and use it appropriately in various mathematical contexts. It's not just about avoiding division by zero; it's about ensuring that the function yields meaningful, real-valued outputs for every input we consider.

The Role of the Numerator

It's important to note that the numerator of a rational function generally does not affect the domain. In our example, $f(x)=\frac{x+6}{(x-7)(x+5)}$, the numerator is $x+6$. If we were to set the numerator to zero, we'd find the roots or zeros of the function (where the function's output is zero):

$$x+6 = 0 \implies x = -6$$

This tells us that $f(-6) = 0$. The value $x = -6$ is perfectly valid as an input and is indeed part of the domain. The existence of zeros in the numerator doesn't create any restrictions on the domain; it simply indicates where the function crosses the x-axis. The only parts of the function that impose restrictions on the domain are those that could lead to undefined operations, with division by zero being the most common culprit for rational functions.

Visualizing the Domain

When we talk about the domain, we are essentially describing the set of all possible $x$-values for which the graph of the function exists. For our function $f(x)=\frac{x+6}{(x-7)(x+5)}$, the graph will exist for all $x$-values except $x = 7$ and $x = -5$. At these $x$-values, there will be vertical asymptotes, which are vertical lines that the graph approaches but never touches. These asymptotes visually represent the points where the function is undefined. If you were to sketch the graph, you would draw vertical lines at $x=7$ and $x=-5$, indicating that the function's graph cannot exist at these specific $x$-coordinates.

Understanding the domain is the first step in analyzing a function's behavior. It provides the boundaries within which we can explore the function's properties. It's like knowing the playing field before you start the game; you need to know the boundaries to play effectively.

Conclusion

In summary, to find the domain of a rational function, we focus solely on the denominator. We set the denominator equal to zero and solve for $x$. The values of $x$ that satisfy this equation are the ones that must be excluded from the set of all real numbers to form the domain. For $f(x)=\frac{x+6}{(x-7)(x+5)}$, the denominator $(x-7)(x+5)$ equals zero when $x=7$ or $x=-5$. Therefore, the domain is all real numbers except -5 and 7. This corresponds to option C.

For further exploration into functions and their properties, you can visit **

Khan Academy's Mathematics section****.