Find the value of the function for the given value of x. See Example 3. ƒ(x)={5 if 02, for x=5.6

Verified step by step guidance

1. Identify which piece of the piecewise function to use based on the given value of x. Since x = 5.6, which is greater than 2, use the second piece of the function: $f(x)=20-3[[2-4x]]$.
2. Substitute x = 5.6 into the expression $2-4x$.
3. Calculate $2-4(5.6)$ to find the value inside the double brackets.
4. Determine the value of the double brackets [[...]], which typically represents the greatest integer function or floor function, meaning you take the greatest integer less than or equal to the value inside.
5. Substitute the result from the double brackets into the expression $20-3[[2-4x]]$ and simplify to find the value of the function.

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 2, and another for values greater than 2. Understanding how to evaluate piecewise functions is crucial for determining the correct output based on the specified input.

Inequalities

Inequalities are mathematical expressions that describe the relationship between values, indicating whether one value is less than, greater than, or equal to another. In the context of the given function, the inequalities (0 < x ≤ 2 and x > 2) dictate which part of the piecewise function to use when evaluating ƒ(x) for a specific value of x.

Function Evaluation

Function evaluation involves substituting a specific input value into a function to determine its output. For the given function ƒ(x), evaluating it at x = 5.6 requires identifying which piece of the function applies and then performing the necessary calculations to find the corresponding output value.