Can the domain and range be 'all real numbers'?

Yes! For many functions, like linear functions (straight lines) or odd-degree polynomials (like $f(x) = x^3$), there are no restrictions on what you can put in, and the graph goes up and down forever. The domain and range would both be $(-\infty, \infty)$.

What if a function has a square root in the denominator?

You have to combine both rules. The inside of the square root must be positive (usually $\geq 0$), but because it's in the denominator, it cannot equal zero. So, you set the expression strictly greater than zero ($> 0$).

Do I use a union symbol (U) in my answers?

You use the union symbol $\cup$ when your domain or range is broken into separate pieces. For example, if a function is valid everywhere except at $x = 0$, the domain is $(-\infty, 0) \cup (0, \infty)$.

How do you find the domain and range of a function?

To find the domain of a function, you look for all the valid input values ( $x$ ) that the function is allowed to use. To find the range, you figure out all the possible output values ( $y$ ) the function can actually produce.

Think of a function like a vending machine. The domain is the type of money the machine accepts—if it only takes quarters, you can't put in a dollar bill. The range is the types of snacks that can come out. You can't get a hot dog if the machine is only stocked with chips and candy. In math, we're just figuring out what numbers are allowed in, and what numbers can come out.

The Domain: Finding Valid Inputs

When looking for the domain, start by assuming it is 'all real numbers.' This means any number will work. Then, play detective and look for math rules that would break the function. If there are no rule-breaking inputs, your domain is indeed all real numbers.

There are two main traps to watch out for. First, you can never divide by zero. If your function has a fraction with $x$ in the denominator, you must exclude any $x$ value that makes the bottom zero. Second, you cannot take the square root (or any even root) of a negative number in real-number algebra. If you have a square root, the expression inside must be greater than or equal to zero.

The Range: Finding Possible Outputs

Finding the range can be a little trickier than finding the domain because you have to imagine what happens to the function after you plug all those valid $x$ values in.

A great way to find the range is to sketch the graph of the function or use a graphing calculator. Look at the lowest point on the $y$ -axis the graph reaches and the highest point. For example, if you have a basic parabola $f(x) = x^2$ , the lowest it ever goes is $0$ , but it goes up forever. So, its range is all numbers greater than or equal to $0$ .

Writing Your Answers

Your teacher will usually want your answer in either inequality notation or interval notation. Inequality notation uses signs like $<$ and $\geq$ . For example, $x \geq 0$ .

Interval notation uses brackets and parentheses. A bracket $[$ or $]$ means the number is included (like $\geq$ or $\leq$ ). A parenthesis $($ or $)$ means the number is not included, and it is always used for infinity ( $\infty$ or $-\infty$ ). For example, $x \geq 0$ is written as $[0, \infty)$ in interval notation.

Where Students Slip Up

A common mistake is forgetting that infinity always gets a parenthesis in interval notation. Infinity isn't a specific number you can reach and 'include,' so you can never put a hard bracket next to it.

Another common slip is swapping domain and range. Just remember: Domain is Input ( $x$ ), Range is Output ( $y$ ). It helps to remember that alphabetically, D comes before R, and $x$ comes before $y$ .

Worked through

Find the domain and range of the function $f(x) = \sqrt{x - 3} + 2$ . Write your answer in interval notation.

First, let's find the domain by looking for input restrictions. We have a square root, and we know the inside of a square root cannot be negative.

So, we set up an inequality: $x - 3 \geq 0$

Add $3$ to both sides: $x \geq 3$

In interval notation, the domain is $[3, \infty)$ .

Next, let's find the range. We know that the square root of any real number is always $0$ or positive. That means the smallest value $\sqrt{x - 3}$ can ever be is $0$ .

If the smallest value of that part is $0$ , then the smallest value of the whole function is: $f(x) = 0 + 2 = 2$

Since the square root can grow infinitely large as $x$ gets larger, the $y$ values will just keep going up from $2$ .

In interval notation, the range is $[2, \infty)$ .

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Where this comes from: OpenStax College Algebra, Chapter 3: Functions · Khan Academy, Algebra 1: Domain and range of a function