Question

\(\operatorname{let} g(x)=1+x-[x]\) and \(f(x)=\mid \begin{array}{cc}-1 & x<0 \\ 0 & x=0 \\ 1 & x>0\end{array}\) Then for all \(\mathrm{x}, \mathrm{f}(\mathrm{g}(\mathrm{x}))\) is equal to (a) \(\mathrm{x}\) (b) 1 (c) \(\mathrm{f}(\mathrm{x})\) (d) \(\mathrm{g}(\mathrm{x})\)

Short answer

The composite function f(g(x)) is equal to 1 for almost all x values, corresponding to option (b). However, there are cases when x is a negative integer where f(g(x)) is not equal to 1. Thus, we cannot definitively say that f(g(x)) is equal to any of the given options.

Step-by-step solution

Analyze the g(x) function

The given function g(x) is 1 + x - [x], where [x] denotes the greatest integer less than or equal to x. This function adds 1 to x but then takes away the greatest integer part of x, effectively leaving only the fractional part of x + 1.

Analyze the f(x) function

The second function, f(x), is piecewise defined as follows: - f(x) = -1 if x < 0 - f(x) = 0 if x = 0 - f(x) = 1 if x > 0 This function checks which of the given intervals the input x belongs to and outputs the corresponding value according to the problem's definition.

Evaluate the composite function f(g(x))

To evaluate f(g(x)), we need to find the value of g(x) and then plug it into the f(x) function. Since g(x) only returns the fractional part of x + 1, we will have the following results for the composite function: - If x is a non-negative integer (x = 0 or x = 1 or x = 2, ...): Then g(x) = x + 1 - [x] = 1. So, f(g(x)) = f(1) = 1 since 1 is greater than 0, which corresponds to option (b) in the given problem. - If x is a negative integer (x = -1 or x = -2 or x = -3, ...): Then g(x) = x + 1 - [x] = -[x]. So, f(g(x)) = f(-[x]), where [x] is positive because x is a negative integer: - If -[x] < 0: f(g(x)) = -1 - If -[x] = 0: f(g(x)) = 0 - If -[x] > 0: f(g(x)) = 1 These outputs do not correspond to any of the given options. - If x is not an integer: Then g(x) = 1 + x - [x] has a fractional part only and is between 0 and 1 (0 < g(x) < 1). In this case, f(g(x)) = 1, which corresponds to option (b) in the problem.

Conclusion

The output for f(g(x)) for all x values is almost always equal to option (b) (1), but there is a case where it is not, specifically when x is a negative integer. As a result, we cannot definitively say that f(g(x)) is equal to any of the given options.

Key concepts

Piecewise Functions

Piecewise functions are like a mathematical puzzle, where each piece of the function applies to different intervals of the input variable. Think of it like a chameleon, changing its behavior based on its surroundings.

For instance, consider the function we have here, referred to as f(x). It is defined differently depending on the value of x. When x is less than zero, f(x) stubbornly settles on the value -1. At exactly zero, it relaxes at a zero. And when x sneaks above zero, the function jumps to 1.

Understanding Boundary Behavior

For piecewise functions, it's crucial to pay attention to the boundaries, those places where the function's rules change. Do we include the boundary point? Do we jump over it? In the case of f(x), we can see that it changes at zero, but includes the value of 0 when x is exactly zero.

Greatest Integer Function

The Greatest Integer Function is like a floor that catches you where you fall – but only if you fall within its integer boundaries. Mathematically, it is denoted as [x] and it represents the largest integer that is less than or equal to x. So if x were 3.7, the greatest integer function would say 'You're 3!'.

Where Function Floors Taper Off

This

Fractional Part of a Number

The fractional part of a number is the 'younger sibling' to the greatest integer function, representing everything that remains after the integer part has been accounted for. You can symbolize it as x - [x], and it whispers, 'Here's what got left behind after taking the integer part away'.

For the number 3.7, while the greatest integer function securely holds onto the 3, the fractional part playfully grabs the 0.7. This is exactly what our function g(x) does by expressing 1+x-[x]: it computes the underlying fraction that remains when the integer part is subtracted from the sum of 1 and x.

In the realm of mathematical analysis, grasping these concepts of the greatest integer and fractional parts sets a solid footing for understanding functions and how they behave under different conditions. And this understanding becomes particularly useful when dealing with the more complex concept of composite functions, where fun unctions layer upon each other to create new, interesting behaviors – much like how f(g(x)) operates in our example.