Problem 4 - Calculus: One and Several Variables

Question

Set \(f(x)=2 x^{2}-3 x+1\) and \(g(x)=x^{2}+1 / x\) Calculate the indicated value. $$\left(\frac{f}{g}\right)(1)$$

Step-by-Step Solution

Answer

The value of \( \left(\frac{f}{g}\right)(1) \) is 0.

Step 1: Evaluate the function f at x = 1

To find \( f(1) \), substitute \( x = 1 \) into the function \( f \), thus \( f(1)=2(1)^{2}-3(1)+1=2-3+1=0 \).

Step 2: Evaluate the function g at x = 1

Similarly, to find \( g(1) \), substitute \( x = 1 \) into the function \( g \), thus \( g(1)=(1)^{2}+1 / 1=1+1=2 \).

Step 3: Evaluate the Ratio

Upon finding the values for \( f(1) \) and \( g(1) \), we then find the ratio \(\frac{f(1)}{g(1)} = \frac{0}{2} = 0 \).

Key Concepts

Function Evaluation

Function evaluation is the process of determining the output value of a function for a specific input value of its variable. In simpler terms, it's like asking what result you get when you "plug in" a number into a function. For the functions given, like \( f(x) = 2x^2 - 3x + 1 \) and \( g(x) = \frac{x^2 + 1}{x} \), finding the value when \( x = 1 \) involves substituting 1 for every \( x \) in the equations.

• Start by replacing \( x \) with 1 in both functions.
• Calculate each term separately to avoid mistakes.
• Combine the calculated terms to find the final result for that specific input.

In our example with \( f(x) \), substitute 1 into the expression to get \( f(1) = 2(1)^2 - 3(1) + 1 = 0 \). Similarly, for \( g(x) \), substitute to find \( g(1) = \frac{1^2 + 1}{1} = 2 \). A solid understanding of function evaluation is crucial for more advanced calculus concepts as well.

Rational Functions

Rational functions are a category of functions defined as the ratio of two polynomials. They can be expressed in the form \( R(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) \) is not zero.

• They often appear in calculus problems related to ratios or divided quantities.
• It's crucial to properly evaluate both the numerator and the denominator at a specific point before attempting to simplify the expression.
• If the denominator evaluates to zero at a point, it indicates the function may not be defined there.

In our exercise, the function \( \left(\frac{f}{g}\right)(x) \) is a rational function where we take \( f(x) \) as the numerator and \( g(x) \) as the denominator. To evaluate \( \left(\frac{f}{g}\right)(1) \), we evaluate \( f(1) \) and \( g(1) \) separately, then divide as \( \frac{0}{2} = 0 \). Understanding this concept is essential when dealing with mathematical expressions involving division or quotients.

Substitution Method

The substitution method is widely used in algebra and calculus to simplify complex expressions by replacing a variable with its value or another expression. This approach allows you to focus on simpler calculations or analyses per component.

• Identify the variable to replace.
• Substitute it with the given value or another expression.
• Solve the resulting expression step-by-step.

In this problem, we applied the substitution method by replacing \( x \) with 1 in both \( f(x) \) and \( g(x) \). For \( f(x) = 2x^2 - 3x + 1 \), after substitution, we calculated each term: \( 2(1)^2 \), \( -3(1) \), and added 1. This gave us \( f(1) = 0 \). For \( g(x) = \frac{x^2 + 1}{x} \), substitution produced a straightforward calculation: \( g(1) = 2 \). Mastery of the substitution method is a key technique that simplifies solving and evaluating functions in calculus.