Problem 31 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate the function \(f\) at the indicated values \(f(-3), f(2), f(-a),-f(a), f(a+h)\). \(f(x)=|x-1|-|x+1|\)

Step-by-Step Solution

Answer

Evaluate the function with the given substitutions for each part.

Step 1: Evaluate at x = -3

First, substitute \(x = -3\) into the function:\[f(-3) = |-3-1| - |-3+1|\]This simplifies to:\[f(-3) = | -4 | - | -2 | = 4 - 2 = 2\]

Step 2: Evaluate at x = 2

Now, substitute \(x = 2\) into the function:\[f(2) = |2-1| - |2+1|\]This simplifies to:\[f(2) = |1| - |3| = 1 - 3 = -2\]

Step 3: Evaluate at x = -a

Substitute \(x = -a\) into the function:\[f(-a) = |-a-1| - |-a+1|\]This simplifies to:\[f(-a) = |-a-1| - |1-a|\]

Step 4: Evaluate at -f(a)

First calculate \(f(a)\) substituting \(x = a\) into the function:\[f(a) = |a-1| - |a+1|\]Then evaluate \(-f(a)\):\[-f(a) = -(|a-1| - |a+1|)\]

Step 5: Evaluate at x = a+h

Substitute \(x = a+h\) into the function:\[f(a+h) = |a+h-1| - |a+h+1|\]This results in:\[f(a+h) = |a+h-1| - |a+h+1|\]

Key Concepts

Absolute Value Functions

Absolute value functions are a type of mathematical expression that involves the magnitude of a number or expression, without considering its sign. Mathematically, we denote absolute value with vertical bars, like this: \(|x|\). Evaluating absolute value means determining how far the number is from zero on a number line, irrespective of direction.
For example:

• The absolute value of 5 is 5, simply put, \(|5| = 5\).
• Similarly, the absolute value of -5 is also 5, which can be written as \(|-5| = 5\).

In the original exercise, absolute value functions play a crucial role in expressing the function \(|x-1| - |x+1|\). These absolute value expressions must be evaluated first before any subtraction or other operations can be performed. It's important to handle these with care, especially when dealing with negative numbers, to avoid mistakes.

Function Substitution

Function substitution refers to the process of replacing a variable in a function with a given expression or value. In essence, it's about plugging a specific value or expression into the function's formula.
Consider the function \(f(x) = |x-1| - |x+1|\). To evaluate this function at any specific value, say \(x = -3\), we substitute -3 in place of x:

• This will transform the function into \(f(-3) = |-3-1| - |-3+1|\) and requires further simplification to determine the final value.

The entire process involves carefully replacing and computing the function for each specific value or expression provided, like \(x = 2, x = -a, -f(a),\) and \(x = a+h\). Each substitution leads to a new functional expression that needs to be resolved.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations like addition, subtraction, multiplication, and division. They are the foundation of algebra and allow us to create symbolic representations of mathematical relationships.
In the function provided, \(f(x) = |x-1| - |x+1|\), the expression inside each set of absolute value bars represents part of an algebraic expression. These expressions need to be simplified, respecting the rules of operation inside the absolute value bars first.
For example, in evaluating \(f(-3)\), the relevant algebraic expressions are \(-3-1\) and \(-3+1\). Once simplified through the absolute value operation, they influence the overall outcome of the expression when substituted back into the function.

Function Evaluation Steps

The process of function evaluation involves systematic steps to determine the value of a function at given points. Here’s a simplified guide to function evaluation summarized from the exercise:

• Step 1: Identify the function form and the value at which it needs evaluation, say \(f(x)\) at \(x = c\).
• Step 2: Substitute the value into the function, replacing x with c in the expression.
• Step 3: Simplify the expression, paying particular attention to operations inside absolute value bars, if they exist.
• Step 4: Solve any remaining operations to get the final result for that specific value.

These steps can be repeated for different values or expressions, such as \(-a, -f(a),\) and \(a+h\), ensuring the function is correctly evaluated each time. By following these steps, you enhance the accuracy of your solutions and ensure consistency across different substitutions.