Problem 39 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate the function at each specified value of the independent variable and simplify. $$f(x)=\frac{|x|}{x}$$ (a) \(f(9)\) (b) \(f(-9)\) (c) \(f(t)\)

Step-by-Step Solution

Answer

So, when evaluated, we get these results: (a) \(f(9)=1\), (b) \(f(-9)=-1\), and (c) \(f(t)=1\) if \(t>0\) and \(f(t)=-1\) if \(t<0\).

Step 1: Evaluate f(9)

We replace \(x\) with 9 in the function \(f(x)\). As 9 is a positive number, its absolute value is equal to its own value. So, \(f(9)=\frac{|9|}{9}\)=\(\frac{9}{9}=1\).

Step 2: Evaluate f(-9)

Meanwhile, we replace \(x\) with -9 in the function \(f(x)\). The absolute value of a negative number is its value without the negative sign, so here, -9 becomes 9. So, \(f(-9)=\frac{|-9|}{-9}\)=\(\frac{9}{-9}=-1\). In the case of negative input values, the function returns -1.

Step 3: Evaluate f(t)

The problem becomes more challenging when variable t is given as an input. The input could theoretically be any number, and demanding absolute value over a generic variable is tricky because we're not sure whether the variable represents a positive or negative number. As a result, the answer here will be a piecewise function. For \(t>0\), \(f(t)=\frac{|t|}{t}=1\) , and for \(t<0\), \(f(t)=\frac{|t|}{t}=-1\).

Key Concepts

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction.
The notation for absolute value is two vertical bars surrounding the number, like this: \( |x| \). Whether the number is negative or positive, its absolute value is always positive or zero.
For example,

• The absolute value of 5 is 5, written as \( |5| = 5 \).
• The absolute value of -5 is also 5, written as \( |-5| = 5 \).
• The absolute value of 0 is simply 0, written as \( |0| = 0 \).

This property is useful when evaluating expressions that involve absolute values, especially when the goal is to simplify the expression to its most basic form.

Piecewise Functions

Piecewise functions are functions defined by different expressions based on specific intervals of the input variable.
This means that the function "chooses" its expression depending on the value of the input. Let's consider the function in the exercise:\[f(t) = \begin{cases} 1, & \text{if } t > 0 \-1, & \text{if } t < 0\end{cases}\]This function evaluates differently based on whether the input, \(t\), is greater than or less than zero.

• For values of \(t\) that are positive, the output is 1.
• For values of \(t\) that are negative, the output is -1.

Piecewise functions are incredibly versatile, as they allow for different behavior based on the input, making them especially useful in real-world contexts where conditions change.

Independent Variable

In the context of functions, the independent variable is the variable that you, the experimenter, can control or choose values for. It is often denoted by \(x\) in equations.
In the given exercise, the independent variable is denoted as \(x\), \(9\), \(-9\), and \(t\).
With the function \(f(x) = \frac{|x|}{x}\), we evaluate the expression for different values of \(x\) or \(t\), treating them as inputs, and observe how the output is affected.

• The chosen value directly influences how the function behaves, such as whether it returns a positive or negative output, or switches between different expressions in a piecewise function.
• Understanding how changes in the independent variable affect the dependent variable (the output) is crucial for analyzing functions and mathematical relationships.

Mastering the concept of an independent variable is essential for solving a wide variety of mathematical problems, as it sets the foundation for understanding how functions work.