Problem 7
Question
Deal with the greatest integer function of Example \(7,\) which is given by the equation \(y=[x]\). Compute the following values of the function: $$[-4 / 3]$$
Step-by-Step Solution
Verified Answer
Answer: \([-4/3] = -2\).
1Step 1: Identify the input value
The input value for the greatest integer function is \(-4/3\).
2Step 2: Find the nearest integer less than or equal to the input value
We need to determine the largest integer that is less than or equal to \(-4/3\). Since \(-4/3\) is between -1 and -2, the greatest integer that is less than or equal to \(-4/3\) is -2.
3Step 3: Compute the function value
The greatest integer function \([x]\) for input \(-4/3\) is equal to the largest integer less than or equal to the input value, which we found to be -2. Therefore, \([-4/3] = -2\).
Key Concepts
Piecewise FunctionsFloor FunctionStep Functions
Piecewise Functions
Piecewise functions are a type of function composed of multiple sub-functions, each of which applies to a certain interval of the domain. In other words, the function behaves differently based on the input value. An easy way to understand piecewise functions is to imagine a chameleon changing colors to blend into various sections of a multicolored background; each color represents a different 'piece' of the function with its own rule.
For example, a function might be defined by one formula for negative inputs and another for positive inputs. Thinking about applying this to real-life situations, piecewise functions can model scenarios like tax brackets, where income up to a certain level is taxed at one rate and income above that level at another. With piecewise functions, it's important to keep track of the domain for each 'piece' to ensure the correct part of the function is used for each value of the variable.
For example, a function might be defined by one formula for negative inputs and another for positive inputs. Thinking about applying this to real-life situations, piecewise functions can model scenarios like tax brackets, where income up to a certain level is taxed at one rate and income above that level at another. With piecewise functions, it's important to keep track of the domain for each 'piece' to ensure the correct part of the function is used for each value of the variable.
Floor Function
The floor function, also known as the greatest integer function, is a type of step function that takes a real number and gives back the greatest integer that's less than or equal to that number. Imagine walking up the stairs; whenever you're on a step, the floor function would give the number of the step you're currently on. Mathematically, it's represented as \( [x] \) where \( x \) is a real number.
For instance, the floor of \( 3.9 \) is \( 3 \) because \( 3 \) is the largest integer not greater than \( 3.9 \) . Similarly, the floor of \( -2.1 \) is \( -3 \) because \( -3 \) is the largest integer that is less than \( -2.1 \). This function is incredibly useful in programming and mathematics for rounding down numbers and finding integer solutions to various problems.
For instance, the floor of \( 3.9 \) is \( 3 \) because \( 3 \) is the largest integer not greater than \( 3.9 \) . Similarly, the floor of \( -2.1 \) is \( -3 \) because \( -3 \) is the largest integer that is less than \( -2.1 \). This function is incredibly useful in programming and mathematics for rounding down numbers and finding integer solutions to various problems.
Step Functions
Step functions are mathematical functions that jump from one value to the next without taking on any values in between. Think of them like the levels in a tiered wedding cake; there's no gradual slope—just distinct levels. Mathematically, each 'jump' between function values can be seen on a graph as a series of flat, horizontal line segments, resembling steps, hence the name.
In the context of the greatest integer function, the graph forms a series of steps that ascend or descend, depending on the direction of the input values. Step functions are useful in situations where quantities change abruptly at specific intervals. For example, shipping rates that increase at certain weight thresholds can be represented by a step function, as the rate jumps to the next price level once a weight limit is surpassed.
In the context of the greatest integer function, the graph forms a series of steps that ascend or descend, depending on the direction of the input values. Step functions are useful in situations where quantities change abruptly at specific intervals. For example, shipping rates that increase at certain weight thresholds can be represented by a step function, as the rate jumps to the next price level once a weight limit is surpassed.
Other exercises in this chapter
Problem 7
Refer to these three functions: $$ \begin{aligned} f(x) &=\sqrt{x+3}-x+1 \\ g(t) &=t^{2}-1 \\ h(x) &=x^{2}+\frac{1}{x}+2 \end{aligned} $$ In each case, find the
View solution Problem 7
Find \((f g)(x),(f / g)(x),\) and \((g / f)(x)\) $$f(x)=x+5, \quad g(x)=x-5$$
View solution Problem 8
Use a calculator and the Horizontal Line Test to determine whether or not the function \(f\) is one-to-one. $$f(x)=.1 x^{3}+.005 x+1$$
View solution Problem 8
Find \((f g)(x),(f / g)(x),\) and \((g / f)(x)\) $$f(x)=\sqrt{x^{2}-1}, \quad g(x)=\sqrt{x-1}$$
View solution