Problem 47 - Precalculus with Limits

Question

In Exercises 37-52, evaluate the function at each specified value of the independent variable and simplify. \(f(x) = |x|/x\) (a) \(f(2)\) (b) \(f(-2)\) (c) \(f(x-1)\)

Step-by-Step Solution

Answer

For the function \(f(x) = |x|/x\), \(f(2) = 1\), \(f(-2) = -1\), and for \(f(x-1)\), if \(x \geq 1\) then \(f(x-1) = 1\) else if \(x < 1\) then \(f(x-1) = -1\).

Step 1: Evaluate \(f(2)\)

To find \(f(2)\), substitute \(2\) into the function: \(f(2) = |2| / 2 = 2/2 = 1\). So, \(f(2) = 1\).

Step 2: Evaluate \(f(-2)\)

To find \(f(-2)\), substitute \(-2\) into the function: \(f(-2) = |-2| / -2 = 2/ -2 = -1\). So, \(f(-2) = -1\).

Step 3: Evaluate \(f(x-1)\)

To find \(f(x-1)\), substitute \(x-1\) into the function: \(f(x-1) = |x-1| / (x-1)\). However, this function needs to be broken down into cases due to the absolute value: If \(x-1 \geq 0\), the function simplifies as: \(f(x-1) = (x-1)/(x-1) = 1\), and if \(x-1 < 0\), the function simplifies as: \(f(x-1) = -(x-1) /(x-1) = -1\). Thus, \(f(x-1) = 1\) when \(x \geq 1\) and \(f(x-1) = -1\) when \(x < 1\).

Key Concepts

Absolute Value Function

The absolute value function is a cornerstone in mathematics, defined as the distance of a number from zero on the number line. It's always non-negative.
When you see the absolute value symbol \( |x| \), it means:

• if \( x \geq 0 \), then \( |x| = x \)
• if \( x < 0 \), then \( |x| = -x \)

For example, \(|2| = 2\) because 2 is positive. Conversely, \(|-2| = 2\) because it's the distance from zero, ignoring the negative sign.
Working with absolute values involves understanding this transformation, which helps in evaluating expressions like \(|x|/x\) correctly when the variable changes.

Function Evaluation

Function evaluation involves substituting a specific value for the variable in the function. It allows us to find out what the function equals for particular inputs.
Consider the function \(f(x) = \frac{|x|}{x}\). To evaluate \(f(2)\), replace \(x\) with 2:\[f(2) = \frac{|2|}{2} = \frac{2}{2} = 1\]Similarly, for \(f(-2)\), replace \(x\) with -2:\[f(-2) = \frac{|-2|}{-2} = \frac{2}{-2} = -1\]Each step involves plugging in the chosen number and simplifying, ensuring you follow the rules of absolute values along the way.

Simplifying Expressions

Simplifying expressions is essential to make functions more comprehensible. Let's break down by handling specific cases based on the sign of the variables.
For \(f(x-1) = \frac{|x-1|}{x-1}\), we need to consider:

• If \(x - 1 \geq 0\), then \(|x-1| = x-1\) and \(f(x-1) = \frac{(x-1)}{(x-1)} = 1\)
• If \(x - 1 < 0\), then \(|x-1| = -(x-1)\) and \(f(x-1) = \frac{-(x-1)}{(x-1)} = -1\)

Through this simplification, the function is understood based on conditions: it outputs 1 when \(x \geq 1\) and -1 when \(x < 1\).
This process illustrates how context and conditions influence the function's outcome, emphasizing why breaking down into cases is crucial.