In Exercises 33–38, express the function, f, in simplified form. Assume that x can be any real number._______f(x) = √81(x-2)²

Verified step by step guidance

1. Identify the expression inside the square root: \(81(x-2)^2\).
2. Recognize that \(81(x-2)^2\) can be rewritten as \((9(x-2))^2\) because \(81 = 9^2\).
3. Apply the property of square roots: \(\sqrt{a^2} = |a|\).
4. Simplify \(\sqrt{(9(x-2))^2}\) to \(|9(x-2)|\).
5. Express the function \(f(x)\) as \(f(x) = 9|x-2|\).

Key Concepts

Square Root Function

The square root function, denoted as √x, is the inverse of squaring a number. It returns the non-negative value that, when squared, gives the original number. In the context of the function f(x) = √81(x-2)², understanding how to simplify square roots is essential, particularly recognizing that √(a²) = |a|, which affects the output based on the value of x.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form while maintaining equivalence. This includes combining like terms, factoring, and applying properties of exponents and roots. In the given function, simplifying √81(x-2)² requires recognizing that 81 is a perfect square and that (x-2)² can be simplified under the square root.

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction, denoted as |x|. When simplifying expressions involving square roots, such as √(x-2)², it is crucial to apply the absolute value, resulting in |x-2|. This concept is vital for accurately expressing the function f(x) in its simplest form.