Problem 1 - Calculus with Concepts in Calculus

Question

Let \(f(x)=2 x^{2}+x-4\) and \(g(x)=3-x^{2} .\) Find the specified values. $$ (f+g)(-1) $$

Step-by-Step Solution

Answer

The value is \(-1\).

Step 1: Understand the Problem

We need to find the value of \((f+g)(-1)\). This means we are looking for the value of the sum of the two functions \(f(x)\) and \(g(x)\) at \(x = -1\).

Step 2: Find \(f(-1)\)

Substitute \(x = -1\) into the function \(f(x) = 2x^2 + x - 4\). Calculate:\[f(-1) = 2(-1)^2 + (-1) - 4\]\[f(-1) = 2(1) - 1 - 4\]\[f(-1) = 2 - 1 - 4\]\[f(-1) = -3\]

Step 3: Find \(g(-1)\)

Substitute \(x = -1\) into the function \(g(x) = 3 - x^2\).Calculate:\[g(-1) = 3 - (-1)^2\]\[g(-1) = 3 - 1\]\[g(-1) = 2\]

Step 4: Calculate \((f+g)(-1)\)

We now add the values of \(f(-1)\) and \(g(-1)\) to find \((f+g)(-1)\).Calculate:\[(f+g)(-1) = f(-1) + g(-1)\]\[(f+g)(-1) = -3 + 2\]\[(f+g)(-1) = -1\]

Step 5: Conclusion

The value of \((f+g)(-1)\) is \(-1\).

Key Concepts

Understanding Polynomial Functions

Polynomial functions are mathematical expressions that involve variables raised to whole number powers. These functions can have one or more terms, and each term comprises a coefficient and a power of the variable. A general polynomial function in variable \(x\) can be expressed as:\[a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\]where \(a_n, a_{n-1}, \ldots, a_0\) are constants known as coefficients, and \(n\) is a non-negative integer representing the highest power, or the degree, of the polynomial. Let's look at our specific functions, \(f(x) = 2x^2 + x - 4\). Here:

• The degree of the polynomial is 2 because the highest power of \(x\) is 2.
• It has three terms: \(2x^2\), \(x\), and \(-4\).
• The coefficients are 2, 1, and -4, respectively.

Understanding polynomials is essential in analyzing the behavior of functions over different intervals.

Techniques for Evaluation of Functions

Evaluating a function means finding the value of the function for a specific input. For a function \(y = f(x)\), we substitute the input value for \(x\) and calculate the resulting expression to find \(y\). In our exercise, we have two functions: \(f(x) = 2x^2 + x - 4\) and \(g(x) = 3 - x^2\).For example, to evaluate \(f(x)\) at \(x = -1\):

• Substitute \(-1\) into \(f(x)\): \(f(-1) = 2(-1)^2 + (-1) - 4\).
• Simplify the expression: \(2 \cdot 1 - 1 - 4 = -3\).
• Thus, \(f(-1) = -3\).

Similarly, evaluate \(g(x)\) at \(x = -1\):

• Substitute \(-1\) into \(g(x)\): \(g(-1) = 3 - (-1)^2\).
• Simplify the expression: \(3 - 1 = 2\).
• Thus, \(g(-1) = 2\).

Being comfortable with plugging in values and simplifying expressions is a critical skill.

The Process of Addition of Functions

Adding functions involves combining two or more functions into one. The addition operation creates a new function expressed as \((f + g)(x)\), where both functions are added together for every input \(x\). This procedure is particularly useful when analyzing the sum of different effects or combining models.For our particular functions, we are finding \((f + g)(x)\), which means:

• For \(x = -1\), calculate \(f(-1)\) and \(g(-1)\).
• We found earlier that \(f(-1) = -3\) and \(g(-1) = 2\).
• Add these results: \((f + g)(-1) = f(-1) + g(-1) = -3 + 2\).
• Thus, \((f + g)(-1) = -1\).

This not only gives you the result of function addition at a certain point but also shows how functions can be dynamically analyzed together.