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Evaluating a function means finding the value of the function for a specific input. Imagine a function as a machine that takes an input, does some processing, and gives an output. For instance, suppose we have the function \(k(x) = \sqrt{x + 1}\). When we put \(x = -10\) into this function, we are basically asking, 'What output does this function give when the input is \(-10\)?' This is known as function evaluation. It's an important concept in mathematics and is widely used in various areas such as physics, engineering, and economics.
To evaluate a function:

• Identify the function and its formula.
• Substitute the given value of \(x\) into the formula.
• Perform the mathematical operations required by the formula.

In our example, we used \(x = -10\) to evaluate the function \(k(x)\), substituting \(-10\) for \(x\) and simplifying as needed.

Imaginary numbers come into play when we take the square root of a negative number. Normally, square roots are defined for non-negative numbers only. However, when we take the square root of a negative number, we get an imaginary number. The imaginary unit is represented by \(i\), where \(i\) is defined as \(\sqrt{-1}\). So, for example, \(\sqrt{-9} = 3i\), since \(\sqrt{9} = 3\) and we have a negative under the square root.
Imaginary numbers are important in a wide range of fields, including engineering, physics, and complex number theory.
They extend the notion of a number line to a complex plane where numbers have both real and imaginary parts.

• The imaginary unit \(i\) satisfies the equation \(i^2 = -1\).
• Any negative square root can be expressed in terms of \(i\); for instance: \(\sqrt{-k} = i\sqrt{k}\).

Square roots are a fundamental concept in mathematics, representing a value that, when multiplied by itself, gives the original number. For example, \(\sqrt{9} = 3\) because \(3 \times 3 = 9\). Square roots of positive numbers are straightforward and always result in a positive value. However, when the number under the square root is negative, this necessitates the use of imaginary numbers, as we saw with \(\sqrt{-9}\).
In our function \(k(x) = \sqrt{x + 1}\), calculating the square root involves taking the value inside the function and simplifying it as much as possible before evaluating the square root.

• Always simplify the expression inside the square root before proceeding.
• Consider the properties of square roots, particularly when dealing with negative numbers.

The substitution method is a critical technique in mathematics, especially when evaluating functions. It involves replacing a variable with a specific value. This method allows us to easily calculate the output of the function for given inputs.
Following these steps can help users effectively use the substitution method:

• Identify the variable to be substituted.
• Replace that variable with the given value.
• Perform the necessary computations to simplify or solve the resulting expression.

For example, in our exercise, we substituted \(-10\) for \(x\) in the function \(k(x) = \sqrt{x + 1}\), transforming it into \(k(-10)\) and then simplifying to get the final result, \(k(-10) = \sqrt{-9} = 3i\).