Domain Of F(x) = √(4 - 3x): A Step-by-Step Guide

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Hey guys! Today, we're diving into a common math problem: finding the domain of a function. Specifically, we'll be tackling the function f(x) = √(4 - 3x). If you've ever wondered what a domain is or how to find it, you're in the right place. We'll break it down step by step, so it's super easy to understand. So, grab your calculators and let's get started!

What is a Function's Domain?

Before we jump into the problem, let's quickly recap what a function's domain actually is. In simple terms, the domain of a function is the set of all possible input values (x-values) for which the function produces a valid output (y-value). Think of it like this: a function is a machine, and the domain is the list of ingredients you can feed into it without breaking it. In other words, the domain consists of all the real numbers that you can plug into the function without causing any mathematical issues. These issues typically arise when we encounter situations like dividing by zero or taking the square root of a negative number. Understanding the domain is crucial for analyzing the behavior and properties of functions in various mathematical contexts.

For instance, consider the function f(x) = 1/x. We can plug in almost any number for x, but there's one big exception: zero. If we try to divide by zero, we get an undefined result. So, zero is not in the domain of this function. Similarly, if we have a square root function, like our example today, we can't take the square root of a negative number (at least, not in the realm of real numbers). This restriction will be key to solving our problem.

Determining the domain of a function is a fundamental step in mathematical analysis. It helps us understand the function's behavior, identify potential issues, and accurately graph the function. By carefully considering the types of operations involved in a function and the values that could lead to undefined results, we can successfully find the domain. With a solid grasp of the domain, we can confidently explore further aspects of the function, such as its range, intercepts, and overall behavior. So, let's keep this definition in mind as we move on to our specific example. It's the foundation for everything else we'll be doing!

Identifying Potential Restrictions

Now that we know what a domain is, let's look at our function: f(x) = √(4 - 3x). Can you spot any potential issues? Remember, we need to find the x-values that will give us a valid output. The big thing to notice here is the square root. Square roots have a very important rule: you can't take the square root of a negative number and get a real number. So, whatever is under the square root sign (in this case, 4 - 3x) must be greater than or equal to zero. This is our key restriction, and it's going to guide us to the solution.

Functions involving square roots require special attention when determining their domains. The expression inside the square root must be non-negative to yield a real number output. This restriction arises from the definition of the square root function within the real number system. If the expression inside the square root is negative, the result would be an imaginary number, which is not considered part of the real number domain. Therefore, to find the domain of a square root function, we need to identify the values of the variable that make the expression inside the square root greater than or equal to zero. This typically involves setting up an inequality and solving for the variable.

Other types of functions also have their own specific restrictions that can affect their domains. For example, rational functions (functions with a variable in the denominator) have the restriction that the denominator cannot be zero, as division by zero is undefined. Logarithmic functions have the restriction that the argument (the expression inside the logarithm) must be positive. Similarly, trigonometric functions like tangent and cotangent have restrictions related to the angles where they are undefined (e.g., tangent is undefined at π/2 + nπ, where n is an integer). Recognizing these potential restrictions is crucial for accurately determining the domain of any function. By carefully considering the operations involved in the function and the values that could lead to undefined results, we can successfully identify the boundaries of the function's domain.

Setting Up the Inequality

Okay, let's translate our restriction into math. We know that the expression under the square root, 4 - 3x, must be greater than or equal to zero. So, we can write this as an inequality:

4 - 3x ≥ 0

This inequality is the heart of our problem. It tells us exactly which x-values will work in our function. Now, all we need to do is solve it!

Inequalities are mathematical statements that compare two expressions using symbols such as greater than (>), less than (<), greater than or equal to (≥), and less than or equal to (≤). Solving an inequality involves finding the range of values for the variable that make the inequality true. The process of solving inequalities is similar to solving equations, but there are a few important differences to keep in mind. One key difference is that multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality sign. For example, if we have the inequality -x > 2, multiplying both sides by -1 would give us x < -2. It's crucial to remember this rule to avoid making errors when solving inequalities. Additionally, when dealing with compound inequalities (inequalities that combine two or more inequalities), we need to solve each inequality separately and then find the intersection or union of their solution sets, depending on the specific compound inequality.

Setting up the correct inequality is a critical step in finding the domain of functions with restrictions. The inequality directly reflects the conditions that must be satisfied for the function to produce valid outputs. For square root functions, the expression inside the square root must be non-negative, leading to an inequality where the expression is greater than or equal to zero. For rational functions, the denominator cannot be zero, so we set up an inequality where the denominator is not equal to zero. For logarithmic functions, the argument must be positive, leading to an inequality where the argument is greater than zero. By accurately translating these restrictions into mathematical inequalities, we can effectively solve for the domain of the function. The inequality serves as a roadmap, guiding us through the algebraic steps needed to determine the permissible values of the variable.

Solving the Inequality

Alright, let's solve 4 - 3x ≥ 0. Our goal is to isolate x on one side of the inequality. Here's how we can do it:

  1. Subtract 4 from both sides:

    -3x ≥ -4

  2. Divide both sides by -3. Important: When we divide by a negative number, we need to flip the inequality sign:

x ≤ 4/3 ```

And that's it! We've solved the inequality. This tells us that x must be less than or equal to 4/3 for the function to be defined.

Solving inequalities requires a careful application of algebraic principles, with a key distinction from solving equations: the reversal of the inequality sign when multiplying or dividing by a negative number. This rule is essential to maintain the correctness of the solution set. The process typically involves isolating the variable on one side of the inequality by performing operations on both sides, such as addition, subtraction, multiplication, and division. Each operation must be applied equally to both sides to preserve the inequality. After isolating the variable, the solution is expressed as a range of values that satisfy the inequality. This range can be represented graphically on a number line or written in interval notation.

When solving inequalities, it's helpful to visualize the solution set on a number line. This visual representation can provide a clear understanding of the values that satisfy the inequality and the boundaries that define the solution. For example, if the solution to an inequality is x < 3, we can draw a number line and shade the region to the left of 3, indicating that all values less than 3 are included in the solution. Similarly, if the solution is x ≥ -2, we shade the region to the right of -2, including -2 itself. Number lines are especially useful when dealing with compound inequalities, as they allow us to easily identify the intersection or union of the solution sets. Moreover, when dealing with inequalities involving absolute values, solving them requires considering two separate cases, as the expression inside the absolute value can be either positive or negative. Solving each case separately and combining the solutions yields the final solution set for the inequality.

Expressing the Domain

So, we know that x ≤ 4/3. How do we write this as the domain of the function? There are a couple of common ways to do it:

  • Inequality Notation: We've already got this! It's just x ≤ 4/3.
  • Interval Notation: This is a more compact way to write the domain. It uses brackets and parentheses to show the range of values. In this case, the domain is (-∞, 4/3]. The parenthesis next to -∞ means that negative infinity is not included (since it's not a specific number), and the square bracket next to 4/3 means that 4/3 is included in the domain.

Expressing the domain of a function is a crucial step in mathematical analysis, as it provides a clear and concise representation of the set of permissible input values for the function. There are several common notations used to express the domain, each with its own advantages and conventions. Understanding these notations is essential for effectively communicating mathematical ideas and solutions.

Inequality notation, as we've seen, directly states the conditions that the variable must satisfy. For example, if the domain consists of all values greater than 2 and less than or equal to 5, we can express this using inequality notation as 2 < x ≤ 5. Interval notation, on the other hand, uses intervals defined by endpoints to represent the domain. Parentheses indicate that the endpoint is not included in the domain, while square brackets indicate that the endpoint is included. Using our previous example, the interval notation for the domain would be (2, 5]. Interval notation is particularly useful for representing domains that consist of multiple intervals or include infinity. Set-builder notation is another method for expressing the domain, which uses set notation to define the set of permissible values. For example, the domain consisting of all real numbers x such that x ≠ 0 can be expressed in set-builder notation as {x | x ∈ ℝ, x ≠ 0}. Each notation offers a unique way to convey the domain, and the choice of notation often depends on the specific context and the preferences of the mathematician or educator.

Putting it All Together

Let's recap what we did. We started with the function f(x) = √(4 - 3x) and wanted to find its domain. We knew that the expression under the square root had to be greater than or equal to zero. So, we set up the inequality 4 - 3x ≥ 0, solved it to get x ≤ 4/3, and then expressed the domain in both inequality notation (x ≤ 4/3) and interval notation ((-∞, 4/3]).

By understanding and applying these techniques, you can confidently determine the domain of a wide range of functions, including those with square roots, rational expressions, and other restrictions. Practice is key to mastering these concepts, and with consistent effort, you'll become proficient in identifying the domain of any function you encounter. Remember to always consider the specific restrictions associated with the type of function you're working with, and translate those restrictions into mathematical inequalities. Solving these inequalities will lead you to the valid range of input values, which defines the domain.

Finding the domain of a function is a foundational skill in mathematics, with far-reaching applications in calculus, algebra, and other areas of study. The domain provides essential information about the function's behavior and limitations, enabling us to analyze its properties and interpret its graphical representation accurately. Mastering the techniques for determining the domain not only enhances your mathematical problem-solving abilities but also deepens your understanding of mathematical concepts. So, embrace the challenge, practice regularly, and you'll find yourself confidently navigating the world of functions and their domains!

Why is the Domain Important?

You might be thinking,