Problem 31 - College Algebra

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

\(f(-3) = 2\), \(f(2) = -2\), \(f(-a) = |a+1| - |a-1|\), \(-f(a) = -( |a-1| - |a+1| )\), \(f(a+h) = |a+h-1| - |a+h+1|\).

Step 1: Evaluate f(-3)

Substitute \(x = -3\) into the function \(f(x) = |x-1| - |x+1|\). Calculate each absolute value separately: \(|-3-1| = 4\) and \(|-3+1| = 2\). The function evaluates to \|4 - 2 = 2\|.

Step 2: Evaluate f(2)

Substitute \(x = 2\) into the function \(f(x) = |x-1| - |x+1|\). Calculate the absolute values: \(|2-1| = 1\) and \(|2+1| = 3\). The function evaluates to \|1 - 3 = -2\|.

Step 3: Evaluate f(-a)

Substitute \(x = -a\) into the function \(f(x) = |x-1| - |x+1|\). Calculate the absolute values: \(|-a-1| = |a+1|\) and \(|-a+1| = |a-1|\). Substitute back into the function: \|f(-a) = |a+1| - |a-1|\|.

Step 4: Evaluate -f(a)

First, find \(f(a)\) by substituting \(x = a\) into \(f(x) = |x-1| - |x+1|\): \|f(a) = |a-1| - |a+1|\|. Now negate the expression to find \-f(a)\|, which gives \-f(a) = -( |a-1| - |a+1| )\|.

Step 5: Evaluate f(a+h)

Substitute \(x = a+h\) into the function \(f(x) = |x-1| - |x+1|\). Compute \(|(a+h)-1| = |a+h-1|\) and \(|(a+h)+1| = |a+h+1|\). The expression becomes \|f(a+h) = |a+h-1| - |a+h+1|\|.

Key Concepts

Absolute Value Function

The absolute value function is a crucial concept in algebra that helps measure the distance of a number from zero on a number line. This function is denoted as \(|x|\), where \(x\) is a real number. The absolute value of any number is always non-negative because it's simply the magnitude of that number, without considering its sign.
For instance:

• \(|3| = 3\) because 3 is already non-negative.
• \(|-3| = 3\) because the negative sign is disregarded, leaving just the magnitude.

In the function given in the exercise, \(f(x) = |x-1| - |x+1|\), the absolute value operation is applied to \(x-1\) and \(x+1\). This means we first find how far these expressions are from zero and then perform subtraction on these absolute values. Understanding this concept is fundamental when evaluating and dealing with absolute value functions.

Substitution

Substitution is a method used in algebra to solve problems by replacing variables with specific numbers or expressions. It's a crucial step in function evaluation that simplifies the process of calculation.
In the exercise, substitution is used effectively to evaluate the function \(f\) at specific values. The process goes as follows:

• Replace \(x\) with the given number, such as -3 or 2, in \(f(x) = |x-1| - |x+1|\).
• Calculate the absolute values for the substituted expressions.
• Evaluate the entire expression to find the result.

Substitution not only aids in solving equations but also helps in understanding how variations in input values affect the function's outcome. When practiced regularly, substitution becomes an invaluable tool for tackling more complex algebraic expressions.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (like addition and multiplication). They are a core component of algebra that allows us to represent real-world problems mathematically.
The function in the exercise, \(f(x) = |x-1| - |x+1|\), is a prime example of an algebraic expression, encompassing:

• Variables: \(x\) represents any real number that we substitute into the function.
• Operations: Involves subtraction between two absolute values.
• Functions: Includes the absolute value function which affects how values are calculated.

Mastering algebraic expressions is key to understanding more advanced topics and solving a variety of mathematical problems. They form the language of algebra, enabling us to describe trends, relationships, and changes succinctly.

College Algebra

College Algebra is a higher-education level course that builds upon the basics of algebra you've learned in high school. It delves into more complex topics and enhances problem-solving skills necessary for advanced mathematics and related fields.
Concepts in college algebra often include:

• Functions and their properties, such as domain and range.
• Complex numbers and their operations.
• Polynomial equations and inequalities.
• Application of algebra in real-world contexts.

In this exercise, understanding how to evaluate functions like \(f(x) = |x-1| - |x+1|\) is crucial as it involves multiple college algebra concepts. As you progress in college algebra, you learn to manipulate and analyze these expressions, preparing you for calculus and beyond. Embracing college algebra opens doors to various career paths in science, engineering, and technology that rely heavily on mathematical proficiency.