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When we talk about evaluating a function, we mean finding the output for given input values. In multivariable calculus, functions often have more than one input variable. The function provided, \(g(x_1, x_2, x_3) = x_1 \sqrt{x_2 x_3}\), is a function of three variables, \(x_1\), \(x_2\), and \(x_3\).

To evaluate the function, you need to substitute the specific values of \(x_1\), \(x_2\), and \(x_3\) as given. This process might seem straightforward, but it is crucial because missing or misplacing values would lead to incorrect outcomes.

• Identify all the variables in the given function.
• Plug in the numbers provided for each respective variable.
• Follow mathematical operations as per the function's definition.

With proper substitution, you transform an abstract formula into a computable expression, making the complex output concrete and understandable.

The substitution method is a staple technique in evaluating functions, especially in calculus. It involves replacing each variable within a function with its corresponding numerical value.

For the function \(g(x_1, x_2, x_3) = x_1 \sqrt{x_2 x_3}\), the given point is \((1, 2, 1)\). This tells us:

• \(x_1 = 1\)
• \(x_2 = 2\)
• \(x_3 = 1\)

Using these values, you substitute them directly into the function:

1. Replace \(x_1\) in the function with 1.
2. Replace \(x_2\) with 2 and \(x_3\) with 1.
3. This turns \(g(x_1, x_2, x_3)\) into an expression you can solve: \(1 \times \sqrt{2 \times 1}\).

Substitution is not just about inserting values randomly. It maintains the function's integrity while personalizing it for specific data points, making this method invaluable in problem-solving scenarios.

The square root calculation is an essential mathematical operation, often seen in multivariable functions. It involves finding a number that, when multiplied by itself, gives the original number.

In the function \(g(x_1, x_2, x_3) = x_1 \sqrt{x_2 x_3}\), after substituting \(x_2 = 2\) and \(x_3 = 1\), you find the expression inside the square root as \(2 \times 1 = 2\).

Now, you have the task of evaluating \(\sqrt{2}\). This expression is the square root of 2. Although \(\sqrt{2}\) does not yield a whole number, it's approximately 1.414. However, in most mathematical contexts, retaining it as \(\sqrt{2}\) is preferable for precision.

• Calculate the product inside the square root.
• Find the square root of the calculated product.
• Use the result in further calculations.

Always remember: the square root maintains part of the mathematical symmetry that is crucial in many calculus problems, offering exactness and clarity in your computations.