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The substitution method in function evaluation is a straightforward approach. It involves replacing variables in the function with specific values. This allows us to compute the function's output for different inputs.

Here's how it's done in simple steps:

• Identify the variable to substitute into the function.
• Replace the variable with the specified value or expression.
• Simplify the expression if needed to find the function's output.

For example, consider the function \( f(x) = \sqrt{3x - 1} \). To evaluate \( f(a) \), substitute \( x = a \) to get \( f(a) = \sqrt{3a - 1} \). By following these steps, you can determine how changing the input affects the output of the function.

The square root function is a type of function that involves the square root of an expression. It is often written as \( f(x) = \sqrt{x} \). This function has a few critical characteristics:

• It only outputs real numbers for non-negative input values (i.e., \(x \geq 0\)).
• The output of the function generally increases as the input becomes larger.
• The function is undefined for negative inputs in the real number system.

In the given function \( f(x) = \sqrt{3x - 1} \), the expression inside the square root \( 3x - 1 \) should be non-negative for the function to be defined in the realm of real numbers. If the expression inside the square root is negative, like in \( f\left(\frac{1}{2}\right) \), the function becomes undefined. Thus, understanding this aspect is crucial when dealing with square root functions.

The domain of a function refers to all the possible input values (usually \( x \)) that allow the function to yield real outputs. Being aware of the domain is crucial since it tells us for which values the function is properly defined.

For functions involving square roots, the domain is especially limited. The expression inside the square root must be zero or positive; otherwise, the function will not work in the real number system. For the function \( f(x) = \sqrt{3x - 1} \), find the domain by setting up the inequality:

• Set \( 3x - 1 \geq 0 \).
• Solve for \( x \): \( 3x \geq 1 \Rightarrow x \geq \frac{1}{3} \).

Thus, the domain of this function is all \( x \) such that \( x \geq \frac{1}{3} \). Understanding the concept of a function's domain helps in correctly evaluating and analyzing it without encountering undefined expressions.