Find the value of the function for the given value of x. See Example 3. ƒ(x)={3 if 04, for x=6.2

Verified step by step guidance

1. Identify which piece of the piecewise function applies for the given value of \( x = 6.2 \). Since \( 6.2 > 4 \), use the second piece of the function: \( f(x) = 10 - 2 \lfloor 5 - x \rfloor \).
2. Substitute \( x = 6.2 \) into the expression inside the floor function: calculate \( 5 - 6.2 \).
3. Evaluate the floor function \( \lfloor 5 - 6.2 \rfloor \), which means finding the greatest integer less than or equal to the result from step 2.
4. Multiply the floor value by \( -2 \) and then add 10 according to the function definition: \( 10 - 2 \times \text{floor value} \).
5. Write the final expression for \( f(6.2) \) after substitution and simplification, but do not calculate the numeric value.

Key Concepts

Piecewise Functions

A piecewise function is defined by different expressions depending on the input value's interval. Understanding how to evaluate such functions requires identifying which part of the function applies to the given input and then using the corresponding formula.

Floor Function (Greatest Integer Function)

The floor function, denoted by [[x]], returns the greatest integer less than or equal to x. For example, [[6.2]] = 6. This concept is essential when evaluating expressions involving floor functions within piecewise definitions.

Evaluating Functions at Specific Points

To find the value of a function at a specific x, determine which part of the function's domain x belongs to, then substitute x into the corresponding expression. This process ensures accurate evaluation, especially for piecewise functions.