Domain Of F(x) = Sqrt(14-2x): Interval Notation

Hey guys! Today we're diving deep into the world of functions, specifically tackling how to find the domain of the function $f(x)=\sqrt{14-2 x}$ and express it using interval notation. This is a super common topic in mathematics, especially when you're first getting your feet wet with algebra and precalculus. Don't sweat it if it seems a bit tricky at first; we'll break it down step by step, making sure you understand every little piece. So, grab your notebooks, maybe a comfy snack, and let's get this math party started!

Understanding What 'Domain' Really Means

Alright, let's kick things off by really understanding what we mean by the 'domain' of a function. Think of a function like a machine. You put something in (an input, usually represented by $x$), and the machine gives you something out (an output, usually $f(x)$ or $y$). The domain of a function is simply the set of all possible inputs that the function can accept and still produce a valid output. It's like asking, "What numbers can I plug into this function without breaking it or getting a nonsensical answer?" In the context of finding the domain of the function $f(x)=\sqrt{14-2 x}$, we need to figure out which values of $x$ will give us a real number as an output.

There are a few common scenarios that can restrict the domain of a function. These include:

1. Division by zero: You can't divide by zero in math, so if your function has a denominator, you need to make sure that denominator never equals zero. For example, in $g(x) = 1/x$, the domain excludes $x=0$.
2. Even roots of negative numbers: This is exactly what we're dealing with in our problem! You can't take the square root (or any even root, like a fourth root or sixth root) of a negative number and get a real number as a result. If you do, you end up with imaginary numbers, which are usually outside the scope of standard domain problems unless specified otherwise. So, for a function like $f(x) = \sqrt{\text{something}}$, the "something" inside the square root must be greater than or equal to zero.
3. Logarithms of non-positive numbers: You can't take the logarithm of zero or a negative number. The argument of a logarithm must be strictly positive.

Since our function, $f(x)=\sqrt{14-2 x}$, involves a square root, our main concern is making sure the expression inside the square root isn't negative. We want to find all the $x$'s that make $14-2x$ a non-negative number. This means $14-2x$ must be greater than or equal to zero. Easy peasy, right? Let's get to solving that inequality!

Solving the Inequality to Find the Domain

Okay, guys, now for the nitty-gritty of finding the domain of the function $f(x)=\sqrt{14-2 x}$. As we established, the expression under the square root, which is $14-2x$, cannot be negative. In mathematical terms, this means:

$14 - 2x \ge 0$

Our goal now is to isolate $x$ and figure out what values of $x$ satisfy this condition. It's a pretty straightforward linear inequality, so let's tackle it.

First, we want to get the term with $x$ by itself. We can do this by subtracting 14 from both sides of the inequality:

$14 - 2x - 14 \ge 0 - 14$

This simplifies to:

$-2x \ge -14$

Now, we need to get $x$ completely alone. To do this, we have to divide both sides by -2. Here's a crucial rule to remember when working with inequalities: whenever you multiply or divide both sides by a negative number, you must flip the direction of the inequality sign. So, that $\ge$ sign becomes a $\le$ sign.

$\frac{-2x}{-2} \le \frac{-14}{-2}$

And when we simplify this, we get:

$x \le 7$

So, what does this tell us? It means that any value of $x$ that is less than or equal to 7 will make the expression $14-2x$ non-negative, and therefore, will result in a real number when plugged into our square root function. This is the core of finding the domain of the function $f(x)=\sqrt{14-2 x}$.

Think about it: if $x=7$, then $14 - 2(7) = 14 - 14 = 0$. The square root of 0 is 0, which is a real number. Perfect!

If $x=6$, then $14 - 2(6) = 14 - 12 = 2$. The square root of 2 is a real number.

If $x=0$, then $14 - 2(0) = 14 - 0 = 14$. The square root of 14 is a real number.

But what if $x$ is greater than 7? Let's try $x=8$:

$14 - 2(8) = 14 - 16 = -2$. The square root of -2 is not a real number. So, $x=8$ is not in the domain.

This confirms our inequality $x \le 7$ is correct. All numbers less than or equal to 7 are valid inputs for our function.

Expressing the Domain Using Interval Notation

We've done the heavy lifting by solving the inequality $x \le 7$. Now, the final step in finding the domain of the function $f(x)=\sqrt{14-2 x}$ is to write our answer using interval notation. This is just a standardized way to represent a set of numbers on the number line.

Interval notation uses parentheses () and square brackets [] to denote ranges of numbers. Here's the quick rundown:

• A parenthesis () means the endpoint is not included in the interval. This is used for inequalities like < (less than) and > (greater than).
• A square bracket [] means the endpoint is included in the interval. This is used for inequalities like $\le$ (less than or equal to) and $\ge$ (greater than or equal to).

We also use infinity symbols ($\infty$ and $-\infty$). Infinity is a concept, not a number, so it's never included in an interval. Therefore, we always use a parenthesis with infinity symbols.

Our result from the inequality is $x \le 7$. This means $x$ can be any number from negative infinity up to and including 7.

• The lower bound: Since $x$ can be any real number going infinitely far to the left on the number line, the lower bound is negative infinity, represented as $-\infty$. As we just said, we always use a parenthesis with $-\infty$, so it's $(-\infty$.
• The upper bound: $x$ can be any number up to and including 7. Since it includes 7, we use a square bracket. So, the upper bound is $7]$.

Putting it all together, the interval notation for $x \le 7$ is:

$(-\infty, 7]$

This notation reads as "all real numbers from negative infinity up to positive 7, including 7."

So, when asked to find the domain of the function $f(x)=\sqrt{14-2 x}$ and write your answer using interval notation, the answer is $(-\infty, 7]$. You've successfully navigated the constraints of the square root function and expressed your findings in the correct format!

Why Interval Notation is Your Friend

Using interval notation, like we just did for finding the domain of the function $f(x)=\sqrt{14-2 x}$, is super handy in mathematics. Instead of writing out "all real numbers less than or equal to 7" every single time, we can just use $(-\infty, 7]$. It's concise, clear, and universally understood by mathematicians. It helps us quickly visualize the set of allowed inputs on a number line.

Think about visualizing $(-\infty, 7]$ on a number line. You'd start way, way out on the left (representing $-\infty$), draw a line going towards the right, and stop at the number 7. You'd put a solid dot or a filled-in circle at 7 because it's included, and then shade the entire line from $-\infty$ up to that point. This visual representation reinforces what the interval notation means.

Understanding how to determine the domain of functions, especially those involving square roots, is a foundational skill. It's crucial for graphing functions, analyzing their behavior, and solving more complex mathematical problems. Keep practicing these types of problems, and you'll become a pro in no time!

Conclusion

So there you have it, guys! We've successfully tackled finding the domain of the function $f(x)=\sqrt{14-2 x}$. By recognizing that the expression under the square root must be non-negative ($14-2x \ge 0$), we solved the inequality to find $x \le 7$. Finally, we translated this inequality into interval notation, giving us the answer $(-\infty, 7]$. This means that only numbers less than or equal to 7 can be plugged into the function $f(x)=\sqrt{14-2 x}$ to produce a real number output. Keep practicing, and you'll master this in no time! Happy calculating!