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 part of the piecewise function to use based on the given value of x. Since x = 6.2, which is greater than 4, use the second part of the function: $f(x)=10-2[[5-x]]$.
2. Substitute x = 6.2 into the expression $5-x$ to find $5-6.2$.
3. Calculate $5-6.2$ to determine the value inside the double brackets, which represent the greatest integer function (also known as the floor function).
4. Apply the greatest integer function to the result from the previous step. The greatest integer function, $[[a]]$, returns the largest integer less than or equal to $a$.
5. Substitute the result of the greatest integer function back into the expression $10-2[[5-x]]$ and simplify to find the value of $f(6.2)$.

Key Concepts

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this case, the function ƒ(x) has two distinct rules: one for values of x between 0 and 4, and another for values greater than 4. Understanding how to evaluate piecewise functions is crucial for determining the correct output based on the specified input.

Inequalities

Inequalities are mathematical expressions that show the relationship between two values, indicating whether one is less than, greater than, or equal to the other. In the context of the given function, the inequalities (0 < x ≤ 4 and x > 4) help define the intervals for which each piece of the function applies, guiding the evaluation process for different values of x.

Function Evaluation

Function evaluation involves substituting a specific value into a function to find the corresponding output. For the given function ƒ(x), evaluating it at x = 6.2 requires using the appropriate piece of the function that applies to this value, which is determined by the defined intervals. This process is essential for finding the function's value at any given point.