Find The Domain Of F(a)=√(2-8a) In Interval Notation

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To find the domain of the function ${f(a) = \sqrt{2 - 8a}}$, we need to determine the set of all possible values of ${a}$ for which the function is defined. The key consideration here is the square root function. The square root of a number is only defined for non-negative values. Therefore, the expression inside the square root must be greater than or equal to zero. In this case, the expression inside the square root is ${2 - 8a}$. So, we need to solve the inequality:

${ 2 - 8a \geq 0 }$

This inequality will give us the range of values for ${a}$ that make the function ${f(a)}$ real and defined. We will solve this inequality step by step to find the domain and then express the solution in interval notation.

Step-by-Step Solution

1. Isolate the term with ${a}$:

   To start, we want to isolate the term ${-8a}$. We can do this by subtracting 2 from both sides of the inequality:

   ${ 2 - 8a - 2 \geq 0 - 2 }$

   This simplifies to:

   ${ -8a \geq -2 }$

2. Divide by the coefficient of ${a}$:

   Next, we need to get ${a}$ by itself. We can do this by dividing both sides of the inequality by -8. It’s important to remember that when we divide or multiply both sides of an inequality by a negative number, we need to reverse the direction of the inequality sign. So, we have:

   ${ \frac{-8a}{-8} \leq \frac{-2}{-8} }$

   This simplifies to:

   ${ a \leq \frac{1}{4} }$

3. Express the solution in interval notation:

   The inequality ${a \leq \frac{1}{4}}$ means that ${a}$ can be any value less than or equal to ${\frac{1}{4}}$. In interval notation, this is represented as:

   ${ (-\infty, \frac{1}{4}] }$

   Here, the parenthesis on the left side indicates that negative infinity is not included in the interval (since infinity is not a number), and the square bracket on the right side indicates that ${\frac{1}{4}}$ is included in the interval.

Detailed Explanation of Each Step

Isolating the Term with ${a}$

The initial inequality is ${2 - 8a \geq 0}$. Our main goal here is to isolate the term that contains the variable ${a}$, which in this case is ${-8a}$. To do this, we perform operations that maintain the balance of the inequality. The first step involves subtracting 2 from both sides. This operation is based on the principle that subtracting the same number from both sides of an inequality does not change the inequality's direction.

By subtracting 2, we eliminate the constant term on the left side, bringing us closer to isolating ${-8a}$. The process is as follows:

${ 2 - 8a - 2 \geq 0 - 2 }$

This simplifies to:

${ -8a \geq -2 }$

This step is crucial because it simplifies the inequality, making it easier to isolate the variable ${a}$ in the next steps. The key concept here is maintaining the balance of the inequality by performing the same operation on both sides.

Dividing by the Coefficient of ${a}$

After isolating the term ${-8a}$, we have the inequality ${-8a \geq -2}$. To solve for ${a}$, we need to divide both sides by the coefficient of ${a}$, which is -8. This is a critical step, and we must be cautious because we are dividing by a negative number. A fundamental rule in dealing with inequalities is that whenever we multiply or divide both sides by a negative number, we must reverse the direction of the inequality sign.

Dividing both sides by -8 gives us:

${ \frac{-8a}{-8} \leq \frac{-2}{-8} }$

Notice that the ${\geq}$ sign has been changed to ${\leq}$ because we divided by a negative number. This step is essential for maintaining the correctness of the solution. Simplifying the fractions, we get:

${ a \leq \frac{1}{4} }$

This inequality tells us that ${a}$ can be any number that is less than or equal to ${\frac{1}{4}}$. This is a clear and concise solution, but to fully understand and use it, we need to express it in interval notation.

Expressing the Solution in Interval Notation

Now that we have the inequality ${a \leq \frac{1}{4}}$, we need to express this solution in interval notation. Interval notation is a way of writing subsets of the real number line. It uses brackets and parentheses to indicate whether endpoints are included in the interval.

In our case, ${a \leq \frac{1}{4}}$ means that ${a}$ can take any value from negative infinity up to and including ${\frac{1}{4}}$. In interval notation, we represent negative infinity as ${-\infty}$, and since infinity is not a number, we always use a parenthesis next to it. The value ${\frac{1}{4}}$ is included in the interval, so we use a square bracket next to it.

Therefore, the interval notation for ${a \leq \frac{1}{4}}$ is:

${ (-\infty, \frac{1}{4}] }$

This notation indicates that the domain of the function ${f(a) = \sqrt{2 - 8a}}$ includes all real numbers from negative infinity up to and including ${\frac{1}{4}}$. Understanding interval notation is crucial for accurately representing the domain and range of functions in mathematics.

Alternative Methods for Solving

While the step-by-step method described above is straightforward, there are alternative approaches to solving the inequality ${2 - 8a \geq 0}$. These methods can provide additional insights and help reinforce the concepts.

Graphical Approach

One alternative method is to use a graphical approach. We can treat the inequality as a linear function and analyze its graph. Let's rewrite the inequality as a function:

${ y = 2 - 8a }$

This is a linear function with a slope of -8 and a y-intercept of 2. To find the domain, we need to determine where this function is greater than or equal to zero.

1. Plot the function:

   The graph of ${y = 2 - 8a}$ is a straight line that slopes downward from left to right. It intersects the y-axis at 2 and the x-axis (where ${y = 0}$) at ${a = \frac{1}{4}}$.

2. Identify the region where ${y \geq 0}$:

   We are interested in the region where the function is greater than or equal to zero. This corresponds to the part of the line that is above or on the x-axis. From the graph, we can see that this occurs when ${a \leq \frac{1}{4}}$.

This graphical approach provides a visual confirmation of our algebraic solution. It helps to understand that the domain of the function corresponds to the values of ${a}$ for which the graph is above or on the x-axis.

Testing Intervals

Another method to solve the inequality is by testing intervals. This approach involves selecting test points from different intervals on the number line and checking whether they satisfy the inequality.

1. Find the critical point:

   The critical point is the value of ${a}$ that makes the expression inside the square root equal to zero. In this case, it is ${a = \frac{1}{4}}$. This point divides the number line into two intervals: ${(-\infty, \frac{1}{4})}$ and ${(\frac{1}{4}, \infty)}$.

2. Select test points:

   Choose a test point from each interval. For example, we can choose ${a = 0}$ from the interval ${(-\infty, \frac{1}{4})}$ and ${a = 1}$ from the interval ${(\frac{1}{4}, \infty)}$.

3. Test the points in the inequality:

   • For ${a = 0}$:

     ${ 2 - 8(0) \geq 0 }$

     ${ 2 \geq 0 }$

     This is true, so the interval ${(-\infty, \frac{1}{4})}$ is part of the solution.

   • For ${a = 1}$:

     ${ 2 - 8(1) \geq 0 }$

     ${ -6 \geq 0 }$

     This is false, so the interval ${(\frac{1}{4}, \infty)}$ is not part of the solution.

4. Include the critical point:

   Since the inequality is ${2 - 8a \geq 0}$, we need to include the critical point ${a = \frac{1}{4}}$ in the solution. This is because the square root of zero is defined.

Thus, the solution is the interval ${(-\infty, \frac{1}{4}]}$, which matches our previous result.

Common Mistakes to Avoid

When finding the domain of functions, especially those involving square roots, it’s easy to make mistakes. Being aware of these common pitfalls can help ensure you arrive at the correct solution.

Forgetting to Reverse the Inequality Sign

One of the most common mistakes occurs when dividing or multiplying an inequality by a negative number. As we discussed earlier, you must reverse the direction of the inequality sign. For example, when solving ${-8a \geq -2}$, dividing both sides by -8 requires changing ${\geq}$ to ${\leq}$. Forgetting this step will lead to an incorrect domain.

Incorrectly Handling Zero

When dealing with square roots, the expression inside the square root must be greater than or equal to zero. This means that the value of the expression can be zero. However, in other contexts, such as rational functions (fractions), the denominator cannot be zero. It’s important to consider these different rules for different types of functions.

Misinterpreting Interval Notation

Interval notation is a specific way of representing intervals on the number line. Using parentheses and brackets correctly is crucial. A parenthesis indicates that the endpoint is not included, while a bracket indicates that it is included. Misinterpreting these symbols can lead to an incorrect representation of the domain.

Not Checking the Solution

After finding a solution, it’s always a good practice to check it. You can do this by selecting a test point from the interval you found and plugging it back into the original inequality. If the inequality holds true, your solution is likely correct. If it doesn’t, you’ll know there’s a mistake somewhere.

Overlooking Domain Restrictions

Functions can have multiple domain restrictions. For example, a function might involve both a square root and a fraction. In such cases, you need to consider all restrictions and find the intersection of the resulting intervals. Overlooking one restriction can lead to an incomplete solution.

Real-World Applications of Domain

Understanding the concept of domain is not just a theoretical exercise; it has practical applications in various real-world scenarios. The domain of a function represents the set of input values for which the function produces a meaningful output. Recognizing and determining the domain is crucial in fields like physics, engineering, economics, and computer science.

Physics

In physics, functions are used to model various phenomena, such as the motion of objects, the behavior of electrical circuits, and the properties of waves. For instance, consider a function that describes the height of a projectile as a function of time. Time cannot be negative in the real world, so the domain of this function would be limited to non-negative values. Similarly, if a function represents the distance an object travels, the domain might be restricted by physical constraints, such as the length of a track or the battery life of a device.

Engineering

Engineers use functions to design and analyze systems and structures. For example, in structural engineering, a function might represent the stress on a beam as a function of the load applied. The domain of this function would be limited by the material properties of the beam and the maximum load it can withstand. Electrical engineers might use functions to model the behavior of circuits, with the domain restricted by voltage limits or component specifications.

Economics

In economics, functions are used to model supply, demand, cost, revenue, and profit. The domain of these functions is often restricted by economic factors. For instance, a demand function might have a domain limited to positive quantities and prices. A cost function might be defined only for production levels that are feasible within the capacity of a factory.

Computer Science

Computer scientists use functions extensively in programming and algorithm design. The domain of a function in a program might be restricted by the data types of the inputs or by the logical constraints of the algorithm. For example, a function that calculates the square root of a number might have a domain limited to non-negative numbers to avoid errors. In database design, functions that retrieve data might have domains restricted by search criteria or access permissions.

Conclusion

In summary, finding the domain of the function ${f(a) = \sqrt{2 - 8a}}$ involves solving the inequality ${2 - 8a \geq 0}$. By isolating the variable ${a}$, we found that ${a \leq \frac{1}{4}}$. Expressing this in interval notation, the domain is ${(-\infty, \frac{1}{4}]}$. This means that the function is defined for all real numbers less than or equal to ${\frac{1}{4}}$. Understanding how to find the domain of functions is crucial for various mathematical and real-world applications.

By following the step-by-step solution, considering alternative methods, and avoiding common mistakes, you can confidently determine the domain of square root functions and other types of functions. This skill is essential for further studies in mathematics and its applications.