Finding The Domain Of F(x) = √(9+x) / (-1+3x) A Step-by-Step Guide

Determining the domain of a function is a fundamental concept in mathematics, crucial for understanding the behavior and limitations of the function. The domain represents the set of all possible input values (x-values) for which the function produces a valid output. In this comprehensive guide, we will delve into the process of finding the domain of the function $f(x) = \frac{\sqrt{9+x}}{-1+3x}$, providing a step-by-step explanation and insightful analysis. This particular function combines two key mathematical elements: a square root and a rational expression, each imposing its own restrictions on the domain. Mastering the techniques to identify these restrictions is essential for anyone studying calculus, precalculus, or related fields. We'll not only find the domain but also discuss the implications of these restrictions on the graph and behavior of the function. Understanding the domain is not just a mathematical exercise; it's about gaining a deeper insight into how functions work and the constraints that govern them. Let's begin our exploration by examining the individual components of the function and their respective domain restrictions.

Understanding the Restrictions

To effectively find the domain of $f(x) = \frac{\sqrt{9+x}}{-1+3x}$, we must first identify the restrictions imposed by the two main components: the square root and the rational expression. The square root component, $\sqrt{9+x}$, introduces the restriction that the expression inside the square root must be non-negative. This is because the square root of a negative number is not defined within the realm of real numbers. Mathematically, this constraint can be represented as $9 + x \geq 0$. Solving this inequality will give us the values of x for which the square root part of the function is valid. On the other hand, the rational expression, $\frac{1}{-1+3x}$, brings in the restriction that the denominator cannot be zero. Division by zero is undefined in mathematics, making it a critical constraint to consider. Therefore, we must ensure that $-1 + 3x \neq 0$. This inequality will help us identify any values of x that would make the denominator zero, and thus, must be excluded from the domain. By addressing these two restrictions systematically, we can accurately determine the set of all permissible x-values for the function. In the subsequent sections, we will meticulously solve these inequalities and combine the results to define the overall domain of the function.

Square Root Restriction: $\sqrt{9+x}$

The first crucial step in determining the domain of our function is to address the square root restriction. As mentioned earlier, the expression inside a square root must be non-negative to yield a real number result. This means that $9 + x$ must be greater than or equal to zero. We can express this as the inequality: $9 + x \geq 0$. To solve this inequality, we need to isolate x on one side. We can do this by subtracting 9 from both sides of the inequality. This gives us: $x \geq -9$. This inequality tells us that the values of x that satisfy the square root restriction are all numbers greater than or equal to -9. In interval notation, this can be represented as $[-9, \infty)$. This interval includes -9 because the inequality is greater than or equal to zero, meaning that -9 is a valid input for the square root part of the function. Understanding this restriction is paramount, as it forms a lower bound for the possible values in the domain. Any value of x less than -9 would result in a negative number inside the square root, leading to an imaginary result, which is not within the scope of real-valued functions. This careful consideration of the square root restriction is a cornerstone of finding the complete domain.

Rational Expression Restriction: $-1 + 3x$

Next, we must consider the rational expression restriction imposed by the denominator of our function, which is $-1 + 3x$. In a rational expression, the denominator cannot be equal to zero, as division by zero is undefined. Therefore, we need to find the values of x that would make the denominator zero and exclude them from the domain. We express this restriction as an inequality: $-1 + 3x \neq 0$. To solve this inequality, we first add 1 to both sides: $3x \neq 1$. Then, we divide both sides by 3 to isolate x: $x \neq \frac{1}{3}$. This result tells us that x cannot be equal to $\frac{1}{3}$. This is a crucial exclusion from our domain, as plugging $\frac{1}{3}$ into the denominator would result in an undefined expression. Unlike the square root restriction which gave us a range of values, this restriction gives us a specific value that must be omitted. In the context of the function's graph, this restriction often corresponds to a vertical asymptote, indicating a point where the function approaches infinity (or negative infinity). By identifying and excluding this value, we ensure that our domain accurately reflects all the valid inputs for the function. This step is just as important as addressing the square root restriction, as both components contribute to the overall domain.

Combining the Restrictions

Now that we've identified the individual restrictions imposed by the square root and the rational expression, the next step is to combine these restrictions to determine the overall domain of the function $f(x) = \frac{\sqrt{9+x}}{-1+3x}$. The square root restriction dictates that $x \geq -9$, which, in interval notation, is $[-9, \infty)$. The rational expression restriction tells us that $x \neq \frac{1}{3}$. To combine these restrictions, we need to exclude the value $\frac{1}{3}$ from the interval $[-9, \infty)$. This means our domain will include all values from -9 up to, but not including, $\frac{1}{3}$, and then continue with all values greater than $\frac{1}{3}$. In interval notation, this combined domain can be expressed as $[-9, \frac{1}{3}) \cup (\frac{1}{3}, \infty)$. The use of parentheses around $\frac{1}{3}$ indicates that this value is not included in the domain, while the square bracket around -9 indicates that -9 is included. The union symbol, $\cup$, combines the two intervals, showing that the domain consists of two separate ranges of values. Visualizing this on a number line can be helpful: imagine a line starting at -9, going all the way up to $\frac{1}{3}$ but with a hole at $\frac{1}{3}$, and then continuing from just after $\frac{1}{3}$ to infinity. This combined domain represents all the possible input values for which the function will produce a valid real number output. Understanding how to combine these restrictions is a key skill in determining the domains of more complex functions.

Expressing the Domain in Interval Notation

Having combined the restrictions, we can now definitively express the domain in interval notation. As established, the domain of the function $f(x) = \frac{\sqrt{9+x}}{-1+3x}$ is the set of all real numbers greater than or equal to -9, except for $\frac{1}{3}$. In interval notation, this is written as $[-9, \frac{1}{3}) \cup (\frac{1}{3}, \infty)$. Let's break down this notation to fully understand it. The square bracket on the left, '[', indicates that -9 is included in the domain. The parenthesis on the right, ')', indicates that $\frac{1}{3}$ is not included in the domain. The interval $[-9, \frac{1}{3})$ represents all real numbers from -9 up to, but not including, $\frac{1}{3}$. The union symbol, $\cup$, is used to combine two intervals. The second interval, $(\frac{1}{3}, \infty)$, represents all real numbers greater than $\frac{1}{3}$, extending to infinity. The parenthesis on the left, '(', again indicates that $\frac{1}{3}$ is not included. By using interval notation, we provide a concise and precise way to represent the domain of the function. This notation is widely used in mathematics and provides a clear understanding of the valid input values for the function. Mastering the use of interval notation is crucial for communicating mathematical ideas effectively.

Conclusion

In conclusion, we have successfully navigated the process of finding the domain of the function $f(x) = \frac{\sqrt{9+x}}{-1+3x}$. By carefully considering the restrictions imposed by both the square root and the rational expression, we determined that the domain is $[-9, \frac{1}{3}) \cup (\frac{1}{3}, \infty)$. This domain represents all real numbers greater than or equal to -9, excluding $\frac{1}{3}$. Understanding how to find the domain of a function is a fundamental skill in mathematics, as it provides insights into the function's behavior and limitations. The combination of different types of expressions, such as square roots and rational expressions, often introduces multiple restrictions that must be carefully considered and combined. The use of interval notation allows us to express the domain clearly and concisely. This comprehensive guide has not only demonstrated the specific steps for this function but also highlighted the underlying principles and concepts applicable to a wide range of functions. By mastering these techniques, you'll be well-equipped to tackle more complex domain problems in calculus and beyond. Remember, the domain is not just a set of numbers; it's a crucial piece of the puzzle in understanding the complete picture of a function.