Problem 45 - Precalculus with Limits

Question

In Exercises 43-50, evaluate the function for the indicated values. \(h(x) = [[x+3]]\) (a) \(h(-2)\) (b) \(h(\frac{1}{2})\) (c) \(h(4.2)\) (d) \(h(-21.6)\)

Step-by-Step Solution

Answer

(a) The value of \(h(-2)\) is 1. (b) The value of \(h(1/2)\) is 3. (c) The value of \(h(4.2)\) is 7. (d) The value of \(h(-21.6)\) is -19.

Step 1: Evaluate h(-2)

Plug -2 into the function. The expression becomes \(h(-2) = [[-2 + 3]] = [[1]]\), and since 1 is an integer, the greatest integer less than or equal to 1 is just 1.

Step 2: Evaluate h(1/2)

Now plug 1/2 into the function. The expression becomes \(h(1/2) = [[1/2 + 3]] = [[3.5]]\). Here, the greatest integer less than or equal to 3.5 is 3.

Step 3: Evaluate h(4.2)

Next, substitute 4.2 into the function. The expression becomes \(h(4.2) = [[4.2 + 3]] = [[7.2]]\). As per the floor function, the greatest integer less than or equal to 7.2 is 7.

Step 4: Evaluate h(-21.6)

Finally, plug -21.6 into the function. The expression becomes \(h(-21.6) = [[-21.6 + 3]] = [[-18.6]]\). According to the floor function, the greatest integer less than or equal to -18.6 is -19.

Key Concepts

Floor Function

The floor function, often denoted by \([ [ x ] ]\), is a mathematical function that takes a real number and rounds it down to the nearest integer. It is also commonly called the "greatest integer function."
This function is essential in various mathematical applications, including computer programming and numerical computations because it simplifies complex numbers by eliminating their fractional components.- For example, the floor of 3.5 is 3, as 3 is the largest integer less than or equal to 3.5.- Similarly, the floor of -18.6 is -19. Despite being a negative number, it is rounded down to the more negative integer.Always remember, no matter what the number is, the floor function will always "round down" to the next lowest integer.

Function Evaluation

Function evaluation is the process of substituting a given number into a function to determine the corresponding output. This is a fundamental operation in mathematics that facilitates understanding how a function behaves at specific points.Consider a simple linear function like \(h(x) = [[x+3]]\). Here, we determine values like \(h(-2)\), \(h(0.5)\), \(h(4.2)\), and \(h(-21.6)\) by substituting these numbers in place of \(x\).- First, plug in the number.- Then, perform any arithmetic operations required.- Finally, apply the floor function to get the greatest integer less than or equal to the result.Each of these steps helps in correctly evaluating the function's outcome. It is also crucial to keep track of signs during arithmetic operations, as negative and positive inputs can drastically alter the floor function's result.

Step Function

A step function is a type of piecewise function that remains constant within each interval of its domain and jumps from one value to another at certain points.
In the context of the exercise, the function \(h(x) = [[x+3]]\) is closely related to what we call a step function because it delivers discrete output values rather than a continuous range of numbers.- This function behaves like a staircase, with each step representing an integer value.- As \(x\) changes, \(h(x)\) "steps" from one integer to the next without ever taking on the intermediate values.By understanding step functions, students can grasp how certain mathematical operations, such as the floor function, interact with their inputs to produce "stepped" results. It is this "stepping" process that constitutes the core of such calculations, making them a pivotal concept in discrete mathematics.