Problem 8 - 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. $$(g \circ f)(1)$$

Step-by-Step Solution

Answer

The result of the function composition \((g \circ f)(1)\) is undefined because it involves a division by zero.

Step 1: Calculate the Value of f(1)

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

Step 2: Replace f(x) with the Computed Value

In the composition of function \((g \circ f)(1)\), replace \(f(x) = 0\) inside the function g(x), so we have \(g(0).\)

Step 3: Calculate the Value of g(0)

Substitute x = 0 into the function g(x). \n \(g(0) = (0)^2 + 1/0 .\) But, division by zero is undefined in mathematics. Therefore, \(g(0)\) is undefined.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions consisting of variables, coefficients, and non-negative integer exponents. These functions are very common in algebra. They can express a wide range of problems and are used across various branches of math.

In the exercise, we have two polynomial functions:

• \( f(x) = 2x^2 - 3x + 1 \)
• \( g(x) = x^2 + \frac{1}{x} \)

The function \( f(x) \) is a simple polynomial function because it involves terms with integer exponents (2, 1, and 0).
However, \( g(x) \) is not a pure polynomial over all real numbers due to the division by \( x \).

Understanding the nature of these functions is crucial for manipulating and evaluating them, especially when dealing with operations like addition, subtraction, and more complex ones like function composition.

Undefined Value

In mathematics, an undefined value emerges when you have expressions with no valid numerical outcome. A typical instance occurs when division by zero happens.

In the solution, division by zero occurs as follows:

• The expression \( \frac{1}{0} \) creates an undefined situation because division by zero is not possible.
• This leads to the conclusion that \( g(0) \) is undefined.

Understanding why a value becomes undefined is crucial, particularly in calculus and analysis, to prevent issues in problem-solving.

In the context of the exercise, knowing when a function or operation results in an undefined value aids in accurate function evaluation and interpretation.

Function Evaluation

Function evaluation involves replacing variables in a function with specific values to obtain a numerical result.

Here, function evaluation is key to finding \((g \circ f)(1)\):

• \( f(1) \) is evaluated by substituting 1 in \( f(x) = 2x^2 - 3x + 1 \), resulting in 0.
• The function composition \( (g \circ f)(1) \) requires plugging \( f(1) \) into \( g(x) \), leading to evaluating \( g(0) \).

This step-by-step process shows how each stage builds upon the previous one.

Correct function evaluation is foundational for problem-solving and allows functions to be systematically transformed and understood across diverse mathematical contexts.Understanding this ensures clarity when dealing with more intricate functions and compositions, as seen with \( (g \circ f)(x) \).