Simplify f(x) square root of 3x-2

Solution:

Step 1:

Rearrange the elements in the expression $f(x)\sqrt{3x-2}$ to $\sqrt{3x-2}f(x)$.

Knowledge Notes:

The problem presented involves simplifying a function that includes a square root. Simplification in mathematics often means to rewrite an expression in a more concise or more elegant form, without changing its value or meaning. In this case, the simplification process does not involve any algebraic manipulation or simplification of the square root itself; rather, it is a simple reordering of the factors in the expression.

The square root symbol $\sqrt{}$ represents the principal square root of a number or expression. For a given positive real number $a$, the principal square root $\sqrt{a}$ is the positive number that, when multiplied by itself, gives $a$. The expression under the square root sign is called the radicand.

In the original problem, $f(x)$ is a function of $x$, and it is multiplied by the square root of the expression $3x-2$. The reordering of the factors does not change the value of the expression, as multiplication is commutative. This means that for any two numbers or expressions $a$ and $b$, the product $ab$ is equal to $ba$.

In the context of functions, reordering factors is often done for stylistic reasons or to prepare the expression for further manipulation, such as integration or differentiation. It's important to note that while reordering terms is permissible in multiplication due to the commutative property, care must be taken when dealing with functions and operations that are not commutative, such as division or subtraction.