Problem 4 - Calculus with Concepts in Calculus

Question

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

Step-by-Step Solution

Answer

\((f/g)(-3) = -\frac{11}{6}\).

Step 1: Understand the expression

We are asked to find \((f/g)(-3)\), which means we need to calculate the ratio of the function \(f(x)\) to \(g(x)\) evaluated at \(x = -3\).

Step 2: Evaluate f(x) at x = -3

Substitute \(-3\) into \(f(x) = 2x^2 + x - 4\):\[ f(-3) = 2(-3)^2 + (-3) - 4 \]\[ = 2(9) - 3 - 4 \]\[ = 18 - 3 - 4 \]\[ = 11 \].

Step 3: Evaluate g(x) at x = -3

Substitute \(-3\) into \(g(x) = 3 - x^2\):\[ g(-3) = 3 - (-3)^2 \]\[ = 3 - 9 \]\[ = -6 \].

Step 4: Calculate (f/g)(-3)

Use the results from Steps 2 and 3 to compute \((f/g)(-3)\):\[ (f/g)(-3) = \frac{f(-3)}{g(-3)} = \frac{11}{-6} \]\[ = -\frac{11}{6} \].

Key Concepts

Polynomial Functions

Polynomial functions are a fundamental part of algebra and calculus. They are expressions that consist of variables raised to whole number powers and are summed up together. For example, in a polynomial function like \(f(x) = 2x^2 + x - 4\), we have terms with different power of \(x\).Polynomials are incredibly useful because they can model a wide range of scenarios:

• Constant terms: These don't change with \(x\) and include numbers like \(-4\) in our function \(f(x)\).
• Linear terms: These involve \(x\) raised to the first power, such as \(x\) in our example.
• Quadratic terms: These are characterized by \(x\) squared, such as \(2x^2\).

Polynomial functions have smooth continuities and can have diverse shapes (like parabolas or waves), depending on the highest power of \(x\). Recognizing the parts of a polynomial helps in analyzing its behavior over different intervals.

Function Evaluation

Function evaluation is the process of finding the output of a function for a particular input. To evaluate, substitute the given value of \(x\) into the function expression, and perform the arithmetic operations as needed.Let's consider the problem where we evaluated \(f(x) = 2x^2 + x - 4\) at \(x = -3\):

• Substitute \(-3\) into \(f(x)\) to get \(f(-3) = 2(-3)^2 + (-3) - 4\).
• Calculate \(2(-3)^2\), which results in \(18\) since \((-3)^2\) is \(9\) and \(2 \times 9 = 18\).
• Then, compute \(18 - 3 = 15\).
• Lastly, perform \(15 - 4\) to get \(11\).

By following these steps, function evaluation becomes a straightforward process. You simplify expressions by handling each operation in a step-by-step manner, ensuring accuracy.

Rational Expressions

A rational expression is the quotient of two polynomial functions. In our exercise, we formed a rational expression by dividing \(f(x)\) by \(g(x)\) and evaluated it at \(x = -3\).Here's a brief guide to working with rational expressions:

• Identify the numerator and the denominator. Here, the numerator is \(f(x) = 2x^2 + x - 4\) and the denominator is \(g(x) = 3 - x^2\).
• Evaluate both the numerator and the denominator at the specified value of \(x\). This means you'll find \(f(-3)\) and \(g(-3)\) separately.
• Substitute these values into the rational expression: \(\frac{f(-3)}{g(-3)}\).
• Simplify the fraction. In our example, \(\frac{11}{-6}\) simplifies to \(-\frac{11}{6}\).

Working with rational expressions requires careful attention to division by zero. Ensure the denominator is not zero for the value of \(x\) being evaluated. This prevents undefined or invalid expressions.