Problem 35 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the function at each specified value of the independent variable. \(f(x)=\llbracket x-1.8 \rrbracket\) (a) \(f(4)\) (b) \(f(3.7)\) (c) \(f(-5.8)\) (d) \(f(-6.3)\)

Step-by-Step Solution

Answer

The function \(f(x)=\llbracket x-1.8 \rrbracket\) evaluated at \(x=4, 3.7, -5.8, -6.3\) gives the results \(f(4)=2\), \(f(3.7)=1\), \(f(-5.8)=-8\), \(f(-6.3)=-9\) respectively.

Step 1: Evaluate Function: Operator Description

The \(\llbracket x \rrbracket\) symbol represents a mathematical floor function. The floor function gives the largest integer less than or equal to a given number.

Step 2: Evaluate Function at \(x=4\)

Plug \(x=4\) into \(f(x)=\llbracket x-1.8 \rrbracket\) to get \(f(4)=\llbracket 4-1.8 \rrbracket= \llbracket 2.2 \rrbracket\). The largest integer less than or equal to 2.2 is 2, so \(f(4)=2\).

Step 3: Evaluate Function at \(x=3.7\)

Similarly, for \(x=3.7\), we get \(f(3.7)=\llbracket 3.7-1.8 \rrbracket = \llbracket 1.9 \rrbracket\). The largest integer less than or equal to 1.9 is 1, so \(f(3.7)=1\).

Step 4: Evaluate Function at \(x=-5.8\)

Substitute \(x=-5.8\) into the function. This gives \(f(-5.8)=\llbracket -5.8-1.8 \rrbracket = \llbracket -7.6 \rrbracket\). The largest integer less than or equal to -7.6 is -8, so \(f(-5.8)=-8\).

Step 5: Evaluate Function at \(x=-6.3\)

Finally, for \(x=-6.3\), we get \(f(-6.3)=\llbracket -6.3-1.8 \rrbracket = \llbracket -8.1 \rrbracket\). The largest integer less than or equal to -8.1 is -9, so \(f(-6.3)=-9\).

Key Concepts

Independent Variable

In mathematics, the independent variable is a crucial concept that often pops up in functions. Simply put, an independent variable is the value that you choose and set, which then affects the outcome of a function. Think of it as the input of a machine where you press different buttons to see various results. In terms of notation, the independent variable is commonly represented by "x." When dealing with functions, you can imagine "x" as the value you control, feeding it into the machine (the function) to see what comes out on the other side (the output).

• It's important to remember that the choice of the independent variable can significantly change the result of a function.
• In our exercise, the independent variable is evaluated at specific points such as 4, 3.7, -5.8, and -6.3.

This concept not only helps in mathematics but also in sciences where you manipulate one variable to observe changes in another.

Integer Approximation

Integer approximation is a powerful technique used in maths that makes dealing with real numbers much simpler. When working with the floor function, we approximate real numbers to integers.
The floor function, represented by \llbracket x - 1.8 \rrbracket in our case, helps in finding the largest integer less than or equal to a real number. It essentially rounds the number down to the nearest whole number, or integer.
For example:

• If you take 2.2, the floor function approximation is "2," as it is the largest integer smaller than 2.2.
• With negative numbers, it also rounds downward, so \(-7.6\) becomes \(-8\).

This type of integer approximation is particularly helpful for simplifying problems where exact decimals aren't necessary or are more cumbersome to work with.

Function Evaluation

Function evaluation is the process of determining the output of a function when given an input value, specifically the independent variable. For example, when you plug "x" into a function like \(f(x)=\llbracket x - 1.8\rrbracket\), you follow certain steps to get your result.
Here's a simple walkthrough:

• Choose the value for the independent variable, "x." This could be any specified number, like 4 or -5.8.
• Input this number into the function equation, replacing "x." For example, \(f(3.7) = \llbracket 3.7 - 1.8 \rrbracket\).
• Calculate the result inside the brackets first. In our case, \(3.7 - 1.8 = 1.9\).
• Use the floor function, which rounds \(1.9\) down to \(1\). Thus, \(f(3.7) = 1\).

Function evaluation is a foundational skill in mathematics, allowing you to apply functions and systems to real-world problems and data.