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Square root functions are a special type of function where the variable is under a square root symbol. This means that we take a number and find what would need to be squared in order to get that number back. For example, if you have the function \( y = \sqrt{x} \), you're finding out what \( y \) should be after taking the square root of \( x \).
This function is often seen in algebra classes because it has unique properties for its domain and range that differ from other functions like linear functions or quadratic functions.

• One key point is that you cannot take the square root of a negative number if you're only using the real number system. So, the input \( x \) must be zero or positive for the function to work.
• Another interesting trait of square root functions is that they grow slower than linear functions. As \( x \) gets larger, the amount by which \( y \) increases gets smaller.

The real number system is made up of all the numbers you normally think of, including integers, fractions, and decimals. It represents all the points on an infinite line where you can always find another number between any two numbers. So you have:

• Whole numbers like 0, 1, 2, etc.
• Fractions like 1/2 or 3/4.
• Irrational numbers like \( \pi \) or the square root of 2.

However, there are limitations to what you can do with real numbers. For example, you can't take the square root of a negative number without moving into the complex numbers, which use the imaginary unit \( i \).
This is why the domain for functions like \( y = \sqrt{x} - 10 \) only includes non-negative numbers because we're staying within the real number system.

Understanding the domain and range of a function is essential. The domain of a function is all the possible input values (x-values), whereas the range consists of all the possible output values (y-values).
For the function \( y = \sqrt{x} - 10 \), the domain comes from the need for \( x \) inside the square root to be zero or positive. Hence, \( x \geq 0 \), which is its domain. This means that you can only substitute zero or positive numbers into \( \sqrt{x} - 10 \).
The range, on the other hand, focuses on the function's resulting values. Since the minimum value of \( \sqrt{x} \) is zero, substituting \( x = 0 \) gives us the smallest possible value of \( y \), which is -10. As \( x \) gets larger, the \( \sqrt{x} \) grows, thus increasing \( y \). Therefore, the range is \( y \geq -10 \).
Knowing how to determine these aspects of a function gives you a strong foundation to describe and manipulate functions accurately.