Find the value of the function for the given value of x. See Example 3. ƒ(x)=[[x/4]], for x=7

Verified step by step guidance

1. Identify the function given: \(f(x) = \left\lfloor \frac{x}{4} \right\rfloor\), where \(\left\lfloor \cdot \right\rfloor\) denotes the floor function, which means rounding down to the nearest integer.
2. Substitute the given value of \(x = 7\) into the function: \(f(7) = \left\lfloor \frac{7}{4} \right\rfloor\).
3. Calculate the division inside the floor function: \(\frac{7}{4} = 1.75\).
4. Apply the floor function to \$1.75\(, which means taking the greatest integer less than or equal to \)1.75$.
5. The result of the floor function is the value of \(f(7)\).

Key Concepts

Function Notation and Evaluation

Function notation, written as ƒ(x), represents a rule that assigns each input x to exactly one output. Evaluating a function means substituting the given value of x into the function's expression and simplifying to find the output.

Floor Function (Greatest Integer Function)

The floor function, denoted by [[x]], returns the greatest integer less than or equal to x. For example, [[3.7]] = 3 and [[-1.2]] = -2. It essentially 'rounds down' any real number to the nearest integer below or equal to it.

Simplifying Algebraic Expressions

Simplifying involves performing arithmetic operations and reducing expressions to their simplest form. In this problem, dividing x by 4 and then applying the floor function requires careful calculation and simplification to correctly evaluate the function.