Problem 49 - Precalculus with Limits

Question

In Exercises 43-50, evaluate the function for the indicated values. \(g(x) = 3[[x - 2]] + 5\) (a) \(g(-2.7)\) (b) \(g(-1)\) (c) \(g(0.8)\) (d) \(g(14.5)\)

Step-by-Step Solution

Answer

(a) \(g(-2.7) = -10\) (b) \(g(-1) = -4\) (c) \(g(0.8) = -1\) (d) \(g(14.5) = 41\)

Step 1: Evaluate g(-2.7)

Firstly, substitute x with -2.7 in the given function: \(g(-2.7) = 3[[-2.7 - 2]] + 5 = 3[[-4.7]] + 5\). Now, the greatest integer function of -4.7 is -5 because -5 is the greatest integer less than -4.7. Therefore, \(g(-2.7) = 3[-5] + 5 = -15 + 5 = -10\)

Step 2: Evaluate g(-1)

Secondly, substitute x with -1: \(g(-1) = 3[[-1 - 2]] + 5 = 3[[-3]] + 5\). The greatest integer function of -3 is -3 itself, so \(g(-1) = 3[-3] + 5 = -9 + 5 = -4\)

Step 3: Evaluate g(0.8)

Next, substitute x with 0.8: \(g(0.8) = 3[[0.8 - 2]] + 5 = 3[[-1.2]] + 5\). The greatest integer less than -1.2 is -2, so \(g(0.8) = 3[-2] + 5 = -6 + 5 = -1\)

Step 4: Evaluate g(14.5)

Lastly, substitute x with 14.5: \(g(14.5) = 3[[14.5 - 2]] + 5 = 3[[12.5]] + 5\). The greatest integer less than or equal to 12.5 is 12, so \(g(14.5) = 3[12] + 5 = 36 + 5 = 41\)

Key Concepts

Piecewise Functions

A piecewise function is a type of function that is defined by different expressions, or 'pieces,' depending on the interval in which its argument, typically denoted as 'x', lies. In other words, the function behaves differently across various segments of its domain. This flexibility allows for the modeling of complex behaviors in mathematical problems.

For example, consider the greatest integer function used in our exercise, which rounds down a number to the nearest whole number less than or equal to it. This action by itself creates a piecewise behavior because the output changes suddenly at the integer boundaries. Therefore, if plotted on a graph, it would appear as a series of steps, hence the name 'step function'. Students often visualize piecewise functions by breaking down the x-axis into intervals and drawing each 'piece' of the function on its respective interval.

Function Evaluation

Function evaluation is the process of finding the output of a function for a particular input. This input is what we typically call 'x'. When you're given a function such as the greatest integer function in the exercise, evaluating it involves substituting the input value into the function and simplifying accordingly.

The greatest integer function notation, denoted by double brackets \( [[x]] \), can be confusing at first. However, once you understand that it acts by rounding down to the nearest integer, function evaluation becomes a straightforward substitution and arithmetic problem. The objective is to always apply the rules of the function consistently for every input value. Doing this correctly is key in precalculus and higher-level mathematics, as it's fundamental in understanding the behavior of functions across their domains.

Precalculus

Precalculus serves as the bridge between algebra and calculus, offering a solid foundation of functions, graph analysis, and limits, which are vital elements in calculus. Working with special functions, such as the greatest integer function, prepares students for the type of rigorous thinking required in calculus.

In this context, precalculus involves operations that range from simple algebraic manipulations to more complex concepts such as function transformations and compositions. The idea is not only to solve equations but to interpret and understand the implications of solutions in a broader mathematical perspective. It's essential for helping students develop an intuition for the continuity and behavior of functions as they progress towards calculus.