Problem 25 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate the indicated function for \(f(x)=x^{2}-1\) and \(g(x)=x-2\) algebraically. If possible, use a graphing utility to verify your answer. $$(f / g)(-5)$$

Step-by-Step Solution

Answer

The solution to \((f / g)(-5)\) for \(f(x) = x^{2} - 1\) and \(g(x) = x - 2\) is \(-24/7\).

Step 1: Define Functions f(x) and g(x)

The problem provides two function definitions: \(f(x) = x^{2} - 1\) and \(g(x) = x - 2\)

Step 2: Find (f/g)(x)

Given that the problem further asks us to find \( (f / g)(-5)\), the first step to solving for this value is to establish the function \( (f / g)(x)\). This can be done by dividing the function \(f(x)\) by \(g(x)\), giving us a new function equal to \( (f / g)(x) = (x^{2} - 1) / (x - 2)\).

Step 3: Substitute (-5) into the Equation

Then, to find \( (f / g)(-5)\), substitute \(-5\) into this established equation. Thus: \((f / g)(-5) = ((-5)^{2} - 1) / ((-5) - 2) = (25 - 1) / -7 = 24 / -7 = -24/7\)

Key Concepts

Understanding Algebra

Algebra is the branch of mathematics that uses symbols and letters to represent numbers and quantities in formulae and equations. It allows us to create expressions and solve them systematically. When dealing with algebraic functions, we often manipulate these symbols according to the rules of arithmetic.

In this problem, functions are expressed using algebraic terms such as \(x^2 - 1\) and \(x - 2\), which are typical in algebra. By understanding algebra, you can solve not only equations but also evaluate expressions by substituting particular values into them, as demonstrated in the given exercise.

Exploring Functions

Functions are a fundamental concept in mathematics, representing a special relationship between input and output values. For each input, there is one unique output. It's like a machine where you put something in and get something out.

In the exercise, we have two functions, \(f(x) = x^2 - 1\) and \(g(x) = x - 2\). Each function is defined by its specific rule or equation. When asked to evaluate \((f/g)(-5)\), we perform operations on these functions, showcasing how they interact and how input values affect outputs. Functions help us model real-world situations and solve complex problems step-by-step.

Delving into Rational Functions

Rational functions are a type of function where one polynomial is divided by another. They can have various features like asymptotes and holes, which occur where the function isn't defined. Understanding rational functions is crucial for calculus and advanced algebra.

In the given problem, we constructed a rational function \((f/g)(x) = \frac{x^2 - 1}{x - 2}\). By inputting \(-5\) into this function, we applied the concept of a rational division of two polynomials. This evaluation often involves checking where denominators become zero, which could invalidate the function at those points, though not an issue here with \(-5\). Rational functions help in understanding complex relationships and behaviors in mathematical models.