Problem 2 - Calculus with Concepts in Calculus

Question

Let \(f(x)=2 x^{2}+x-4\) and \(g(x)=3-x^{2} .\) Find the specified values. $$ (f-g)(2) $$

Step-by-Step Solution

Answer

The value of \((f-g)(2)\) is 7.

Step 1: Understand the Expression

The question asks for the value of \((f-g)(2)\), which means we are finding the value of the function \(f(x) - g(x)\) at \(x = 2\). This is equivalent to calculating \(f(2) - g(2)\).

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

First, substitute \(x = 2\) into the function \(f(x) = 2x^2 + x - 4\).\[f(2) = 2(2)^2 + (2) - 4 = 2(4) + 2 - 4\]Simplify it:\[= 8 + 2 - 4 = 6\]

Step 3: Calculate \(g(2)\)

Next, substitute \(x = 2\) into the function \(g(x) = 3 - x^2\).\[g(2) = 3 - (2)^2 = 3 - 4\]Simplify it:\[= -1\]

Step 4: Calculate \((f-g)(2)\)

Now, use the values calculated in the previous steps to find \((f-g)(2)\):\[(f-g)(2) = f(2) - g(2) = 6 - (-1)\]Simplify it:\[= 6 + 1 = 7\]

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions that consist of variables and constants combined using addition, subtraction, and multiplication, but not division by variables. These functions are expressions made up of terms called monomials. Each monomial is characterized by a coefficient (a constant) and a variable raised to a non-negative integer power.

For example, in the polynomial function given in the exercise, we have two specific polynomials:

• The function \(f(x) = 2x^2 + x - 4\)
• The function \(g(x) = 3 - x^2\)

Here, the term \(2x^2\) in \(f(x)\) is a monomial of degree 2, the term \(x\) is degree 1, and \(-4\) is a constant term. In \(g(x)\), the term \(-x^2\) is a degree 2 term, and the number 3 is a constant. Understanding these components is crucial to working with polynomial functions as they often form the basis of more complex algebraic operations.

Evaluation of Functions

Evaluating a function means finding the value of the function for a particular value of its variable. This is done by substituting the specified value into the function and simplifying the result.

The exercise asked to evaluate

• \(f(2)\), which is found by substituting \(x = 2\) into \(f(x) = 2x^2 + x - 4\), resulting in \(f(2) = 6\)
• \(g(2)\), achieved by substituting \(x = 2\) into \(g(x) = 3 - x^2\), resulting in \(g(2) = -1\)

Through these steps, you can see how substitution and arithmetic come together to determine the results. This approach forms the foundation for understanding more complex operations such as function subtraction or other function operations.

Subtraction of Functions

Subtraction of functions involves finding the difference between two functions for any given value of x. When subtracting, you take one function and subtract the entire expression of the other function. This operation can be symbolically expressed as \((f-g)(x) = f(x) - g(x)\).

In the exercise, this concept was demonstrated when finding \((f-g)(2)\). We calculated it by using the values previously found,

• \(f(2) = 6\)
• \(g(2) = -1\)

Thus,

• \((f-g)(2) = 6 - (-1) = 6 + 1 = 7\)

It is important to carefully apply arithmetic rules, particularly with negative numbers or subtraction operations. This understanding sets the stage for performing more complicated operations and analysis on functions, an essential skill in algebra and calculus.