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The square root function is one of the fundamental functions in mathematics. It involves taking a number and finding another number that, when multiplied by itself, gives the original number. In simpler terms, it's about finding the number that "when squared" returns the given value. For instance, the square root of 9 is 3 because 3 squared (or 3 times 3) equals 9.
In any function involving a square root, it's crucial to ensure that the value under the square root is non-negative because the square root of a negative number is not defined in the set of real numbers; it would be an imaginary number. In our exercise, the expression under the square root is \( \sqrt{x-2} \). Therefore, \( x-2 \) must be equal to or greater than 0, leading us to the inequality \( x \geq 2 \). This is how we derive part of the domain restriction for functions containing square roots.

Logarithms are fascinating and vital concepts in mathematics because they help us deal with very large or very small numbers. The logarithmic function is the inverse of the exponential function. In simpler terms, if you think of exponentiation as multiplying a number by itself a certain number of times, taking a logarithm would find out "how many times" to achieve a certain result.
In any function involving a logarithm, such as \( \log_{5}(10-x) \), the expression inside the logarithm must be positive for it to be defined in real numbers. That's because logarithms of non-positive numbers are undefined in the real number system.
In the context of our problem, \( 10-x \) must be greater than 0, leading to the inequality \( x < 10 \). This is why we have another restriction on the domain of our function, because the logarithm must have a positive argument.

Inequalities are mathematical statements that describe the relative size or order of two objects. Unlike equations, which show equality, inequalities show one side is larger or smaller than the other. Inequalities are crucial in determining the domain of functions, especially when dealing with restrictions like the ones from square roots and logarithms.
In our case, we derived the inequalities \( x \geq 2 \) from the square root and \( x < 10 \) from the logarithm. These inequalities guide us in finding which values of \( x \) are allowable in our function's domain. By combining these conditions, we understand that \( x \) values must satisfy both inequalities simultaneously for the function \( h(x) \) to work correctly. The solutions to these inequalities essentially give us the valid domain of the function.

Interval notation is a concise way of writing the set of all \( x \) values that satisfy an inequality or a combination of inequalities. It uses brackets to describe end values, with parentheses indicating that an endpoint is not included and square brackets showing endpoint inclusion.
In our exercise, after finding the inequalities \( x \geq 2 \) and \( x < 10 \), we combine these to state the domain of the function in interval notation. The bounds are included in square brackets and parentheses as appropriate, resulting in the interval \( [2, 10) \). This means \( x \) can be equal to 2 (included) and can be less than, but never equal to, 10 (not included).
Interval notation is a powerful way to communicate ranges simply and effectively, especially in calculus and algebra.