Problem 31 - College Algebra

Question

In Exercises \(21-32,\) evaluate each function at the given values of the independent variable and simplify. $$f(x)=\frac{x}{|x|}$$ a. \(f(6)\) b. \(f(-6)\) c. \(f\left(r^{2}\right)\)

Step-by-Step Solution

Answer

The evaluations of the function at the given values of the independent variable are: a) f(6)=1 b) f(-6)=-1 c) For \(r\ne0\), f(\(r^{2}\))=1 and for \(r=0\), f(\(r^{2}\)) is undefined.

Step 1: Understanding the Function

The function \(f(x)=\frac{x}{|x|}\) returns 1 for all positive values of x, -1 for all negative values of x and is undefined for x=0.

Step 2: Evaluate f(6)

We substitute x=6 into the function to get \(f(6)=\frac{6}{|6|}=1\)

Step 3: Evaluate f(-6)

Next, substituting x=-6 into the function, \(f(-6)=\frac{-6}{|-6|}=-1\)

Step 4: Evaluate f(r^2)

Likewise, for \(f(r^{2})\), \(r^{2}\) is always non-negative, so \(f(r^{2})=\frac{r^{2}}{|r^{2}|}=1\) if \(r\ne0\). If \(r=0\), the expression is undefined.

Key Concepts

Absolute Value

The absolute value is a fundamental concept in mathematics, representing the distance of a number from zero on the number line, regardless of direction. In notational terms, the absolute value of a number 'x' is written as \( |x| \). For example, both \( |3| \) and \( |-3| \) equal 3, as both points are three units away from zero.
For positive numbers, the absolute value is simply the number itself, while for negative numbers, it’s the positive counterpart of the number. The absolute value of zero is zero, since it’s precisely at the origin of the number line. Importantly, the absolute value function is always non-negative.
In the context of the function \( f(x)=\frac{x}{|x|} \), you’re dividing a number by its absolute value. This operation essentially normalizes the number to 1 or -1, depending on the sign, as seen in the evaluations of \( f(6) \) and \( f(-6) \). Whenever approaching problems with absolute values, it's important to consider the case when the inside of the absolute value is zero, as dividing by zero is undefined, leading to an undefined function value.

Function Simplification

Function simplification is the process of making a function's form as straight-forward as possible by reducing complexity, often to make evaluation or further manipulation easier. Simplifying a function may involve combining like terms, factoring, expanding, or reducing fractions.
In our case, the function \( f(x) \) simplifies the idea of sign determination into a numerical output of 1 or -1. This simplification can make it easier to understand the function’s behavior. Simplified functions are particularly useful when you need to quickly evaluate numerous values of an independent variable, as they allow for a faster and clearer identification of patterns or properties.
Moreover, when simplifying functions, one aims to present the expression in a form that clearly shows the critical features without extra complications. It's a 'less is more' approach that can be especially beneficial in learning environments, aiding in the student's comprehension of the function's behavior.

Independent Variable

In mathematics, an independent variable is a variable that stands alone and isn't affected by other variables you are trying to measure. It's often denoted by 'x' and represents the input of a function, where the output value is determined based on the rule of the function.
When evaluating functions, it's crucial to understand the role of the independent variable, as it drives the value of the dependent variable, or the output. Changing the independent variable’s value allows us to explore the function’s behavior and understand its characteristics.
For instance, in the given function \( f(x) = \frac{x}{|x|} \), 'x' is the independent variable. When we plug different values of 'x' into the function, such as in \( f(6) \), \( f(-6) \) or \( f\left(r^{2}\right) \), the outcomes are a direct result of the independent variable’s value. It’s important to recognize that independent variables can represent not just simple numbers but also expressions, as long as they are substitutable into the function.