Problem 10 - Calculus with Concepts in Calculus

Question

Let \(f(x)=\frac{x-1}{x^{2}+1}\) and \(g(x)=x^{1 / 4}\). Find the specified values. $$ (f-g)(1) $$

Step-by-Step Solution

Answer

\((f-g)(1) = -1\)

Step 1: Understand the Problem

The problem asks us to find \((f - g)(1)\). This means we are looking for the expression \(f(1) - g(1)\). First, we need to find the value of \(f(1)\) and \(g(1)\) and then subtract them.

Step 2: Calculate \(f(1)\)

Substitute \(x = 1\) into the function \(f(x) = \frac{x-1}{x^2+1}\). Calculate it:\[ f(1) = \frac{1-1}{1^2+1} = \frac{0}{1+1} = 0 \]

Step 3: Calculate \(g(1)\)

Substitute \(x = 1\) into the function \(g(x) = x^{1/4}\). Calculate it:\[ g(1) = 1^{1/4} = 1 \]

Step 4: Evaluate \((f-g)(1)\)

Subtract the value of \(g(1)\) from \(f(1)\):\[ (f-g)(1) = f(1) - g(1) = 0 - 1 = -1 \]

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in calculus, where we find the output of a function given a specific input value. Consider the functions given in the exercise:

• f(x) = \(\frac{x-1}{x^2+1}\): a rational function.
• g(x) = x^{1/4}: a root function.

To evaluate these functions, substitute the given input into each formula. For example, to find \(f(1)\), substitute \(x = 1\) into \(f(x)\): \[ f(1) = \frac{1-1}{1^2+1} = \frac{0}{2} = 0 \]Next, to find \(g(1)\), substitute \(x = 1\) into \(g(x)\): \[ g(1) = 1^{1/4} = 1 \]The function evaluation tells us that \(f(1)\) gives 0 and \(g(1)\) gives 1. This step is crucial as it provides the numerical values that will be used in operations like subtraction.

Subtraction of Functions

Subtraction of functions involves computing the difference between the outputs of two functions for the same input. In this exercise, we're asked to subtract \(g\) from \(f\) at \(x = 1\), which means finding the value of \((f-g)(1)\). This is expressed as:\[ (f-g)(x) = f(x) - g(x) \]Upon substituting \(x = 1\):

• Calculate \(f(1) = 0\) as derived from evaluation.
• Calculate \(g(1) = 1\) as derived from evaluation.
• Finally, compute \(f(1) - g(1)\).

The result of subtraction at this specific input is:\[ (f-g)(1) = f(1) - g(1) = 0 - 1 = -1 \]This operation highlights how different functions can be combined or subtracted, providing new insights into the relationships between various functional behaviors.

Step-by-step Problem Solving

Approaching calculus problems with a step-by-step method ensures clarity and simplification of potentially complex operations. Let's break down how it applies to this problem:- **Understand the Problem:** Identify that we need to find \((f-g)(1)\), which involves evaluating two functions at \(x = 1\) and finding the difference between them.- **Evaluate Each Function:** As seen previously, calculate \(f(1)\) and \(g(1)\). This part involves substituting \(x = 1\) into each function and simplifying.- **Combine STEP Results:** We use the evaluations to compute \((f-g)(1)\). By subtracting \(g(1)\) from \(f(1)\), we find: \[ (f-g)(1) = f(1) - g(1) = 0 - 1 = -1 \]Each step builds upon the last, ensuring each part is understood before proceeding. This methodical approach is very effective in calculus to manage tasks systematically and avoid mistakes.