Problem 25 - Precalculus with Limits

Question

In Exercises 17-28, evaluate the indicated function for \(f(x) = x^2 + 1\) and \(g(x) = x - 4\). \((f/g)(5)\)

Step-by-Step Solution

Answer

\((f/g)(5) = 26\).

Step 1: Function Division

Firstly, understand how function division is performed. When you see \((f/g)(x)\), it means that you have to divide the function \(f(x)\) by the function \(g(x)\).

Step 2: Substitute the Formulas

Now, substitute the given functions into the formula: \(f(x) / g(x) = (x^2 + 1) / (x - 4)\).

Step 3: Evaluate at x = 5

Substitute \(x = 5\) into the formula: \((f/g)(5) = ((5)^2 + 1) / ((5) - 4) = (25 + 1) / (5 - 4) = 26.\)

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in mathematics that involves finding the value of a function for a specific input. It's like asking a machine to do its job when given a particular tool, in this case, a number. Imagine you have a function machine, and you feed it an input number. The machine processes this input to produce an output. In our exercise, functions like \( f(x) = x^2 + 1 \) and \( g(x) = x - 4 \) are such machines. You input a value, say 5, and the machine churns out a result. For instance, evaluating \( f(5) \) involves plugging 5 into the function \( f(x) \), yielding \( 25 + 1 = 26 \). This method helps you understand the behavior of functions at particular points.

Rational Functions

Rational functions are a special category of mathematical functions. They are essentially the fraction of two polynomial functions. It's like having a relationship that describes how one thing can be expressed in terms of another using simple ratios. In our context, the function \( (f/g)(x) = \frac{f(x)}{g(x)} \) represents a rational function. Here, \( f(x) = x^2 + 1 \) is the numerator, and \( g(x) = x - 4 \) is the denominator. These functions are key because they appear frequently in algebra and calculus and are used to model various real-world phenomena. Understanding and manipulating these expressions can help build a solid foundation for analyzing complex mathematical relationships. Just remember: the denominator must not be zero, as division by zero is undefined.

Substituting Values

Substituting values involves replacing variables in an expression with specific numbers to evaluate the expression. It's the moment when everything becomes real and we can get a concrete number out of potentially complex equations. For example, in our solution, we substitute \( x = 5 \) into \( \frac{x^2 + 1}{x - 4} \). This means wherever you see \( x \), you replace it with 5. Thus, the expression becomes \( \frac{5^2 + 1}{5 - 4} \). Calculating this, \( 25 + 1 \) gives 26, and \( 5 - 4 \) gives 1, resulting in \( \frac{26}{1} = 26 \). Substitution is an essential skill that allows you to test hypotheses and verify solutions in mathematics.