Finding The Domain Of $f(x)=\sqrt{4x+9}+2$ A Step-by-Step Explanation

Determining the domain of a function is a fundamental concept in mathematics, and it's essential for understanding the behavior and limitations of that function. In this article, we will delve into the process of finding the domain of the function $f(x) = \sqrt{4x + 9} + 2$. We'll break down the steps, explain the underlying principles, and provide a clear understanding of why a specific inequality is used to define the domain.

Understanding the Domain of a Function

Before we dive into the specifics of our function, let's establish a solid understanding of what the domain of a function actually means. Simply put, the domain is the set of all possible input values (often represented by 'x') for which the function produces a valid output (often represented by 'y' or $f(x)$). In other words, it's the range of x-values that we can plug into the function without encountering any mathematical roadblocks.

These roadblocks can arise from several situations, including:

• Division by zero: A fraction with zero in the denominator is undefined.
• Square root of a negative number: In the realm of real numbers, the square root of a negative number is not defined.
• Logarithm of a non-positive number: The logarithm of zero or a negative number is undefined.

For our function, $f(x) = \sqrt{4x + 9} + 2$, the primary concern is the square root. We need to ensure that the expression inside the square root, $4x + 9$, is non-negative.

The Key Inequality: $4x + 9 \geq 0$

Now, let's address the core question: which inequality can be used to find the domain of $f(x)$? The correct answer is $4x + 9 \geq 0$. But why?

The reason lies in the fundamental property of square roots. As mentioned earlier, we cannot take the square root of a negative number within the set of real numbers. Therefore, to ensure that $f(x)$ produces a real output, the expression inside the square root, $4x + 9$, must be greater than or equal to zero.

This inequality, $4x + 9 \geq 0$, mathematically expresses this requirement. It states that the value of $4x + 9$ must be either positive or zero. By solving this inequality, we can determine the range of x-values that satisfy this condition, which in turn defines the domain of the function.

Why Other Options Are Incorrect

Let's briefly examine why the other provided options are not suitable for finding the domain of $f(x)$:

• $\sqrt{4x} \geq 0$: This inequality focuses solely on the term $4x$, neglecting the crucial '+ 9' within the square root of our original function. It doesn't account for the fact that even if $4x$ were negative, the '+ 9' might compensate and make the entire expression under the square root non-negative.
• $4x \geq 0$: Similar to the previous option, this inequality only considers $4x$ and ignores the '+ 9'. It doesn't provide a complete picture of the condition required for the square root to be defined.
• $\sqrt{4x + 9} + 2 \geq 0$: While this inequality is true for all x-values within the domain, it doesn't directly help us find the domain. The expression $\sqrt{4x + 9} + 2$ will always be greater than or equal to 2 as long as $4x+9$ is non-negative. This inequality describes a property of the function's output, not the constraint on its input.

Solving the Inequality and Determining the Domain

Now that we've established that $4x + 9 \geq 0$ is the correct inequality, let's solve it to find the domain of $f(x)$.

1. Isolate the x term: Subtract 9 from both sides of the inequality: $4x \geq -9$

2. Solve for x: Divide both sides by 4: $x \geq -\frac{9}{4}$

Therefore, the solution to the inequality is $x \geq -\frac{9}{4}$. This means that the domain of the function $f(x) = \sqrt{4x + 9} + 2$ is all real numbers greater than or equal to $-\frac{9}{4}$.

In interval notation, we can express the domain as $[-\frac{9}{4}, \infty)$. This notation indicates that the domain includes all numbers from $-\frac{9}{4}$ up to positive infinity, including $-\frac{9}{4}$ itself.

Visualizing the Domain

It's often helpful to visualize the domain on a number line. To represent the domain $x \geq -\frac{9}{4}$, we would draw a closed circle (or a square bracket) at $-\frac{9}{4}$ and shade the line to the right, indicating that all values greater than or equal to $-\frac{9}{4}$ are included in the domain.

Importance of Domain in Function Analysis

The domain of a function is not just an abstract mathematical concept; it's a crucial piece of information for understanding the function's behavior and its applicability to real-world scenarios. Knowing the domain allows us to:

• Identify valid inputs: We can avoid plugging in values that would lead to undefined results.
• Interpret the function's output: The domain helps us understand the range of possible outputs the function can produce.
• Model real-world situations accurately: In applications, the domain often represents physical constraints or limitations on the input variables.

For example, if our function represented the height of an object over time, a negative value in the domain would likely be meaningless since time cannot be negative. Similarly, in a function modeling the population of a species, the domain might be restricted to non-negative integers.

Conclusion

In conclusion, determining the domain of a function is a fundamental skill in mathematics. For the function $f(x) = \sqrt{4x + 9} + 2$, the inequality $4x + 9 \geq 0$ is the key to finding the domain. This inequality ensures that the expression inside the square root remains non-negative, allowing the function to produce real-valued outputs. By solving this inequality, we found that the domain of $f(x)$ is $x \geq -\frac{9}{4}$, or in interval notation, $[-\frac{9}{4}, \infty)$. Understanding and determining the domain is crucial for analyzing the behavior of functions and applying them effectively in various contexts.

By mastering the concept of domain, you gain a deeper understanding of how functions work and their limitations, paving the way for more advanced mathematical explorations.