Problem 97
Question
If \(f(x)=1+x-[x]\), then find the function \(g(x)=\operatorname{sgn}(f(x))\)
Step-by-Step Solution
Verified Answer
\(g(x) = \operatorname{sgn}(f(x)) = \left\{ \begin{array}{ll}
1 & \text{ if } x \text{ is not an integer} \\
0 & \text{ if } x \text{ is an integer}
\end{array}\right.\)
1Step 1: Analyze f(x) for the given definition of [x]
First, let's notice that [x] represents the greatest integer less than or equal to x. As a result, [x] always takes an integer value. Since we need to find the sign of f(x), let's check when f(x) is positive, negative, or zero.
2Step 2: Determine when f(x) is positive
We have the following inequality for the positive values of f(x):
\(f(x) > 0 \Rightarrow 1 + x - [x] > 0\)
Since [x] is always less than or equal to x, we have:
\[x - [x] > 0\]
Now, let's add 1 to both sides of the inequality:
\(1 + x - [x] > 1\)
So, f(x) is positive when x is greater than [x]. This means that f(x) is positive for all x, except when x is an integer.
3Step 3: Determine when f(x) is negative
No matter what the value of x is, f(x) will never be negative because [x] is always less than or equal to x, which means that x - [x] is always greater than or equal to 0. Therefore, f(x) is always positive or zero, but never negative.
4Step 4: Determine when f(x) is zero
f(x) will be zero when:
\(f(x) = 0 \Rightarrow 1 + x - [x] = 0 \Rightarrow x - [x] = -1\)
This occurs only when x is an integer because the difference between x and the greatest integer less than or equal to x is 1.
5Step 5: Write the final function g(x)
Now that we have determined when f(x) is positive, negative, or zero, we can write g(x) as follows:
\[g(x) = \operatorname{sgn}(f(x)) = \left\{ \begin{array}{ll}
1 & \text{ if } x \text{ is not an integer} \\
0 & \text{ if } x \text{ is an integer}
\end{array}\right.\]
So, the function g(x) is equal to 1 when x is not an integer and equal to 0 when x is an integer.
Key Concepts
Greatest Integer FunctionSignum FunctionPiecewise FunctionInteger and Non-Integer Analysis
Greatest Integer Function
The greatest integer function, often denoted as \([x]\), is a function that assigns the greatest integer less than or equal to a given number \(x\).
This means if \(x\) is an integer, \([x] = x\), and if \(x\) is not an integer, \([x]\) becomes the nearest integer less than \(x\).
For example:
The greatest integer function plays a crucial role in analyzing integer and non-integer behavior.
This means if \(x\) is an integer, \([x] = x\), and if \(x\) is not an integer, \([x]\) becomes the nearest integer less than \(x\).
For example:
- \([3.7] = 3\)
- \([-2.3] = -3\)
- \([5] = 5\)
The greatest integer function plays a crucial role in analyzing integer and non-integer behavior.
Signum Function
The signum function, represented as \(\operatorname{sgn}(x)\), provides the sign of a number. It returns three possible values:
We found \(f(x)\) is always non-negative, simplifying \(g(x)\) to return \(1\) when \(f(x) > 0\) and \(0\) when \(f(x) = 0\).
The signum function is especially important in defining behaviors of piecewise functions.
- \(1\) if \(x > 0\)
- \(0\) if \(x = 0\)
- -1 if \(x < 0\)
We found \(f(x)\) is always non-negative, simplifying \(g(x)\) to return \(1\) when \(f(x) > 0\) and \(0\) when \(f(x) = 0\).
The signum function is especially important in defining behaviors of piecewise functions.
Piecewise Function
A piecewise function is defined by different expressions based on the input's value. In this exercise, \(g(x)\) is a piecewise function that considers whether \(x\) is an integer.
It is expressed as:
Piecewise functions allow flexibility by adapting their form based on input conditions, making them highly valuable in mathematical modeling.
It is expressed as:
- \(g(x) = 1\) if \(x\) is not an integer
- \(g(x) = 0\) if \(x\) is an integer
Piecewise functions allow flexibility by adapting their form based on input conditions, making them highly valuable in mathematical modeling.
Integer and Non-Integer Analysis
Analyzing integers and non-integers provides critical insights into functions like \(f(x) = 1 + x - [x]\).
- For non-integer \(x\), \(x - [x] > 0\), so \(f(x) > 0\). This implies that the function behaves differently compared to integer values. - For integer \(x\), however, \(x - [x] = 0\), making \(f(x) = 0\).
These properties are vital in determining outputs of functions like \(g(x)\), showing stark differences in behavior based solely on whether \(x\) is an integer or not.
Understanding this analysis is essential for solving problems involving functions that change based on integer and non-integer inputs.
- For non-integer \(x\), \(x - [x] > 0\), so \(f(x) > 0\). This implies that the function behaves differently compared to integer values. - For integer \(x\), however, \(x - [x] = 0\), making \(f(x) = 0\).
These properties are vital in determining outputs of functions like \(g(x)\), showing stark differences in behavior based solely on whether \(x\) is an integer or not.
Understanding this analysis is essential for solving problems involving functions that change based on integer and non-integer inputs.
Other exercises in this chapter
Problem 95
Show that \(\left[x+\frac{1}{2}\right]-\left[x-\frac{1}{2}\right]=1 \forall x\).
View solution Problem 96
Plot graphs:- i. \(\quad f(x)=[\sin x]\). ii. \(\quad f(x)=\sin ^{-1}[x]\). iii. \(y=\operatorname{sgn}[x]\). iv. \(y=\ln [\sin x]\). v. \(\quad f(x)=\left|e^{\
View solution Problem 98
Show that the function \(f(x)=[x]+[-x]\) has removable discontinuity for integral values of \(x\).
View solution Problem 101
Prove that \(f(x)=\frac{2 x(\sin x+\tan x)}{2\left[\frac{x+2 \pi}{\pi}\right]-3}\) is an odd function.
View solution