Finding The Domain Of F(x) = √(5x - 5) + 1 A Detailed Explanation

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When we delve into the world of functions, a fundamental concept to grasp is the domain of a function. The domain essentially defines the set of all possible input values (often represented as x) for which the function produces a valid output. In simpler terms, it's the range of x-values that you can plug into the function without causing mathematical errors. For the function $f(x) = \sqrt{5x - 5} + 1$, we encounter a specific type of function: a square root function. These functions have a crucial restriction – we cannot take the square root of a negative number within the realm of real numbers. This restriction dictates the inequality we must use to determine the function's domain.

Identifying the Key Restriction: The Square Root

The core of the function $f(x) = \sqrt{5x - 5} + 1$ lies in the square root term, $\\sqrt{5x - 5}$. As mentioned earlier, the square root of a negative number is not defined within the set of real numbers. This is because there is no real number that, when multiplied by itself, results in a negative number. For instance, the square root of -4 is not a real number because there is no real number that, when squared, equals -4. Therefore, to ensure that our function produces real outputs, the expression inside the square root, known as the radicand, must be non-negative. This means it must be either greater than or equal to zero.

Formulating the Inequality: Ensuring a Non-Negative Radicand

Based on the restriction identified above, we can formulate the inequality that governs the domain of our function. The radicand, which is $5x - 5$, must be greater than or equal to zero. This translates directly into the following inequality:

5x−5geq05x - 5 \\geq 0

This inequality is the key to unlocking the domain of the function. It states that the expression $5x - 5$ cannot be negative; it must be either zero or a positive number. Solving this inequality will give us the range of x-values that are permissible inputs for the function.

Solving the Inequality: Determining the Domain

Now, let's solve the inequality $5x - 5 \geq 0$ to find the domain. We'll follow a step-by-step approach:

  1. Isolate the term with x: Add 5 to both sides of the inequality:

    5x−5+5geq0+55x - 5 + 5 \\geq 0 + 5

    5xgeq55x \\geq 5

  2. Solve for x: Divide both sides of the inequality by 5:

    frac5x5geqfrac55\\frac{5x}{5} \\geq \\frac{5}{5}

    xgeq1x \\geq 1

The solution to the inequality is $x \geq 1$. This means that the domain of the function $f(x) = \sqrt{5x - 5} + 1$ consists of all real numbers x that are greater than or equal to 1. In interval notation, we represent this domain as $[1, \infty)$. This interval includes 1 (as indicated by the square bracket) and extends infinitely to the right, encompassing all numbers greater than 1.

Verifying the Domain: Testing Values

To solidify our understanding and ensure the correctness of our domain, let's test a few values. We'll choose values within the domain (x ≥ 1) and outside the domain (x < 1) to observe the function's behavior.

Testing a Value Within the Domain (x = 2)

Let's substitute x = 2 into the function:

f(2)=sqrt5(2)−5+1f(2) = \\sqrt{5(2) - 5} + 1

f(2)=sqrt10−5+1f(2) = \\sqrt{10 - 5} + 1

f(2)=sqrt5+1f(2) = \\sqrt{5} + 1

The result, $\sqrt{5} + 1$, is a real number. This confirms that a value within our determined domain produces a valid output.

Testing a Value Outside the Domain (x = 0)

Now, let's substitute x = 0 into the function:

f(0)=sqrt5(0)−5+1f(0) = \\sqrt{5(0) - 5} + 1

f(0)=sqrt−5+1f(0) = \\sqrt{-5} + 1

Here, we encounter the square root of a negative number, $\\sqrt{-5}$, which is not a real number. This demonstrates that a value outside our domain leads to an undefined result in the real number system.

These tests reinforce the accuracy of our domain calculation. The function is well-defined for all x-values greater than or equal to 1, and it produces non-real outputs for values less than 1.

Why Other Options Are Incorrect

Now, let's analyze why the other inequality options provided are incorrect:

  • A. 5x - 4 ≥ 0: This inequality would lead to a domain of $x \geq \frac{4}{5}$, which is incorrect. While $\frac{4}{5}$ is a real number, it does not satisfy the requirement that the radicand (5x - 5) must be non-negative.
  • B. √(5x - 1) + 1 > 20: This is not an inequality used to find the domain. It's an inequality that could be used to find the range of x values for which the function's output is greater than 20, but it doesn't define the domain.
  • C. 5x - 5 < 0: This inequality describes the values of x for which the radicand is negative, which is precisely what we want to avoid when determining the domain of a square root function.
  • D. 5x - 3 ≥ 0: This inequality would lead to a domain of $x \geq \frac{3}{5}$, which is also incorrect for the same reason as option A. It doesn't ensure that the radicand (5x - 5) is non-negative.

Conclusion: The Importance of the Radicand

In conclusion, the correct inequality to determine the domain of the function $f(x) = \sqrt{5x - 5} + 1$ is $5x - 5 \geq 0$. This inequality stems from the fundamental restriction that the radicand (the expression inside the square root) must be non-negative for the function to produce real outputs. Solving this inequality, we find that the domain of the function is $x \geq 1$, which includes all real numbers greater than or equal to 1. Understanding and applying this principle is crucial for working with square root functions and accurately determining their domains.

By carefully analyzing the function and considering the restrictions imposed by the square root, we can confidently identify the correct inequality and determine the function's domain. This process highlights the importance of understanding the underlying principles of mathematical functions and their limitations.

The correct inequality is:

  • 5x−5geq05x - 5 \\geq 0