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Radical expressions are mathematical expressions that include a radical symbol, which indicates the root of a number. The most common radical expression is the square root, but radicals can also represent cube roots, fourth roots, and so on. The general form of a radical expression is \( \sqrt[n]{x} \), where \( x \) is the radicand (the number under the radical symbol) and \( n \) is the index (which tells us which root we're taking).

In our example, \( \sqrt{y} \) is a radical expression, where \( y \) is the radicand. Radical expressions require careful handling, especially when it comes to simplification. To simplify a radical expression, one must look for perfect squares, cubes, etc., that can be taken out of the radical. For instance, \( \sqrt{4} \) simplifies to \( 2 \) because \( 4 \) is a perfect square, that is, \( 2^2 \) equals \( 4 \) and consequently, the square root of \( 4 \) is \( 2 \) .

• Recognize the radical symbol and index.
• Identify the number under the radical as the radicand.
• Look for perfect power factors to simplify the expression.

Function evaluation is the process of determining the output of a function for a specific input. In mathematical notation, if \( f \) is a function and \( x \) is the input, then \( f(x) \) denotes the value of the function when the input is \( x \) . In other words, it tells us what we get when we replace the function's variable with a given number or expression.

To evaluate our example function \( f(y)=3-\sqrt{y} \) for a specific value of \( y \) , we follow these general steps:

• Substitute the specific value or expression for \( y \) into the function.
• Simplify the radical expression if necessary.
• Perform the arithmetic operations to find the result.

In the step-by-step solution provided, we evaluated \( f(y) \) for \( y = 4 \) , \( y = 0.25 \) , and \( y = 4x^2 \). Each of these evaluations requires substituting the value of \( y \) and simplifying the resulting expression to find the final output of the function.

Square roots are a particular type of radical expression that denote the principal root of a number. A square root of a number \( x \) answers the question: 'What number, when multiplied by itself, will equal \( x \) ?' The square root is written as \( \sqrt{x} \) and is the inverse operation of squaring a number.

For positive real numbers, the square root always has two solutions: a positive root and a negative root, but by convention, when we write \( \sqrt{x} \) we refer to the positive root. For example, \( \sqrt{9} \) equals \( 3 \) and not \( -3 \) because we take the principal, or non-negative, root by default.

• The square root of a perfect square will be an integer (e.g., \( \sqrt{16} = 4 \)).
• The square root of a non-perfect square will be an irrational number (e.g., \( \sqrt{2} \)).
• To simplify, factor the radicand into perfect squares and take the square root of each factor separately.

Working with square roots also means being cautious with negative numbers since the square root of a negative number isn't real in the realm of real numbers and requires the use of complex numbers and the imaginary unit \( i \) . However, in the provided exercise, we are focusing on real number inputs only.