Problem 2 - Calculus: One and Several Variables

Question

Calculate (a) \(f(0),\) (b) \(f(1),(\text { c) } f(-2), \text { (d) } f(3 / 2)\). $$f(x)=\frac{2 x-1}{x^{2}+4}$$

Step-by-Step Solution

Answer

The function evaluates to (a) -1/4, (b) 1/5, (c) -5/4, and (d) 8/25 at x = 0, 1, -2, and 3/2 respectively.

Step 1: Evaluate Function at x = 0

By substituting x = 0 into the function, \(f(0)=\frac{2(0)-1}{(0)^{2}+4}=-\frac{1}{4}\).

Step 2: Evaluate Function at x = 1

By substituting x = 1 into the function, \(f(1)=\frac{2(1)-1}{(1)^{2}+4}=\frac{1}{5}\).

Step 3: Evaluate Function at x = -2

By substituting x = -2 into the function, \(f(-2)=\frac{2(-2)-1}{(-2)^{2}+4}=-\frac{5}{4}\).

Step 4: Evaluate Function at x = 3/2

By substituting x = 3/2 into the function, \(f(3/2)=\frac{2(3/2)-1}{(3/2)^{2}+4}=\frac{2}{\frac{25}{4}}=\frac{8}{25}\).

Key Concepts

Substitution

Substitution is a method used to solve equations or evaluate expressions by replacing variables with specific numbers. It's like filling in the blanks with given numbers. For example, to find the value of a function at a given point, we replace the variable in the function with the given number. This approach simplifies the calculation.
In this exercise, you're asked to calculate \(f(x)\) at different values of \(x\): 0, 1, -2, and 3/2.
The substitution process involves:

• Taking the function \(f(x)=\frac{2x-1}{x^{2}+4}\)
• Replacing \(x\) with each specified value
• Calculating the result to find the function’s value at that point

Substitution helps break down complex expressions into simple arithmetic, making it easier to understand what the function does at specific points.

Rational Function

A rational function is a type of mathematical expression that involves the division of two polynomials. It resembles a fraction but with polynomials. Understanding rational functions is crucial, as they appear often in real-world problems and various fields like physics and engineering.
In the function \(f(x)=\frac{2x-1}{x^{2}+4}\), we have:

• A numerator which is \(2x-1\)
• A denominator \(x^{2}+4\)

The numerator and denominator are both polynomials. Rational functions can take different shapes and forms, depending on their inputs. The behavior can vary significantly when substituting different values for \(x\) due to the structure of the fraction. For instance, rational functions can produce undefined points when a denominator equals zero, though in this case it's always positive.

Algebraic Manipulation

Algebraic manipulation involves modifying mathematical expressions into different forms. This can help simplify expressions or solve them more easily. Using algebraic manipulation is key to understanding how different mathematical structures work and can be transformed.
When evaluating the function \(f(x)=\frac{2x-1}{x^{2}+4}\), several manipulations occur:

• Rewriting \(2x-1\), calculating by multiplying, and possibly subtracting numbers
• Computing \(x^2+4\) by squaring the input and adding a constant

The result is a simplified expression that allows for easy calculation. This algebraic manipulation turns a potentially complex expression into a clear numerical value. It’s a tool that makes comprehending and working with mathematical functions more accessible.