Problem 67 - College Algebra with Modeling and Visualization

Question

Domain and Range If \(f(k)=-6,\) what is the value of \(1 f(k) | ?\)

Step-by-Step Solution

Answer

The value is 6.

Step 1: Substitute the given value

We are given that \(f(k) = -6\). Now, substitute \(-6\) in place of \(f(k)\) in the expression \(|1f(k)|\).

Step 2: Simplify the expression

The expression now becomes \(|1f(k)| = |1(-6)|\). Simplifying inside the absolute value, this turns into \(|-6|\).

Step 3: Apply absolute value

The absolute value of a negative number \(-6\) is its positive counterpart, which means \( |-6| = 6 \). Therefore, \(|1f(k)| = 6\).

Key Concepts

Absolute Value

When you see a number or expression within vertical bars, like \(|x|\), it represents the absolute value. The absolute value tells you how far a number is from zero on the number line, regardless of its direction. This means that absolute values are always non-negative. - For any positive number, \(|a| = a\). - For any negative number, like -6, the absolute value is the positive version: \(|-6| = 6\).Let's consider the statement: "The absolute value measures the size without considering direction." This means it provides the magnitude of the number only. This is especially helpful when you deal with expressions containing negative numbers. When you calculate something like \(|1(-6)|\), you're setting aside the fact that the number is negative. Therefore, knowing how to handle absolute values is essential in math for simplifying expressions and solving equations.

Function Notation

Functions in math are like machines that take inputs and give you outputs. They are often written in function notation, such as \(f(x)\). Here, \(f\) is the name of the function, and \(x\) represents the input.Function notation helps in clearly defining relationships. For instance, when we have \(f(k) = -6\), it tells us that when \(k\) is the input, the output is -6. This notation is wonderful because it organizes information systematically, making it easier to plug in numbers or variables to see results.Think of function notation as a way to label the process and output. It is extremely useful for identifying how a certain variable affects another in mathematical problems. For example, by putting different values into \(f\), you can efficiently determine the corresponding outputs without much hassle. This notation aids in easily detecting patterns and functions.

Algebraic Expressions

Algebraic expressions are like phrases in mathematics. They involve numbers, variables, and operations, all mixed together. For example, in our exercise, we have \(|1f(k)|\), which contains numbers "1" and "-6", a variable "k", and operations like multiplication and absolute value.The beauty of algebraic expressions is in their simplicity in expressing complex mathematical ideas. These expressions can be simplified using rules, such as combining like terms, or performing operations like addition and multiplication. For instance, multiplying 1 by \(f(k)\) simply results in \(f(k)\). Adding the absolute value function, the expression becomes much easier to handle, as seen when we simplify \(|1(-6)|\) to \(|-6|\) and finally to 6.Understanding how to work with algebraic expressions can unlock the ability to solve equations, explore functions, and determine domains and ranges. When tackled step-by-step, these expressions provide a structured way to solve math problems with clarity and precision.