Problem 46 - Thinking Mathematically

Question

Evaluate each function at the given value of the variable. \(f(x)=\frac{|x|}{x}\) a. \(f(5)\) b. \(f(-5)\)

Step-by-Step Solution

Answer

The results are: \(f(5) = 1\), and \(f(-5) = -1\).

Step 1: Evaluate \(f(5)\)

Substitute \(x=5\) in the function: \(f(5)=\frac{|5|}{5}=1\). Thus the function outputs 1 when \(x=5\).

Step 2: Evaluate \(f(-5)\)

Substitute \(x=-5\) in the function: \(f(-5)=\frac{|-5|}{-5}=-1\). Thus the function outputs -1 when \(x=-5\).

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in mathematics that involves replacing the variable in a function with a given number and executing the operations. For example, when we are given the function \( f(x) = \frac{|x|}{x} \), and asked to evaluate \( f(5) \), we substitute \( x \) with 5 in the function. This means we solve the expression by carrying out the operations based on the substituted value.

• Step 1: Substitute \( x = 5 \) into the function \( f(x) = \frac{|x|}{x} \). This becomes \( f(5) = \frac{|5|}{5} \).
• Step 2: Calculate the absolute value of 5, which is 5, then divide 5 by 5 to get 1. Therefore, \( f(5) = 1 \).

Similarly, for \( x = -5 \), we again substitute and evaluate:

• Step 1: Substitute \( x = -5 \) into the function. It becomes \( f(-5) = \frac{|-5|}{-5} \).
• Step 2: The absolute value of -5 is 5. Divide 5 by -5 to get -1. Hence, \( f(-5) = -1 \).

Function evaluation helps in determining the output value of a function for a particular input.

Piecewise Functions

Piecewise functions are unique because they are defined by different expressions based on the input values. These functions can have multiple "pieces" or segments, each applying to a specific part of the domain. A prime example of a piecewise function is the absolute value function, which is expressed as \( |x| \).

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

In our exercise, the function \( f(x) = \frac{|x|}{x} \) can be thought of as piecewise. This is because the expression within \( \frac{|x|}{x} \) changes according to whether \( x \) is positive or negative.

• For \( x > 0 \), the function outputs 1 because \( |x| = x \).
• For \( x < 0 \), it outputs -1, because \( |x| = -x \).

By understanding how piecewise definitions apply, we can easily evaluate functions like these by choosing the correct expression based on the input range.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operations that represent a particular quantity or concept. In dealing with absolute value functions, expressions can sometimes involve signs and absolute values which need specific handling to simplify.
Consider \( f(x) = \frac{|x|}{x} \) again. This is a compact expression that involves both division and an absolute value operation:

• The denominator, \( x \), is straightforward as it keeps the variable in its original form.
• The numerator, \( |x| \), is the absolute value of \( x \), meaning it is always non-negative, regardless of whether \( x \) itself is positive or negative.

When we evaluate this expression for specific values, like 5 or -5, we use the rules of absolute value to simplify:

• Evaluating at 5 means both the numerator and denominator are positive, resulting in a positive output.
• Evaluating at -5 means the numerator becomes positive \( (|-5| = 5) \), and the denominator remains negative, yielding a negative output.

The ability to manipulate and simplify expressions by applying correct rules is crucial in evaluating functions.