Problem 1 - Basic Engineering Mathematics

Question

If \(f(x)=4 x^{2}-3 x+2\) find: \(f(0), f(3), f(-1)\) and \(f(3)-f(-1)\)

Step-by-Step Solution

Answer

\(f(0) = 2\), \(f(3) = 29\), \(f(-1) = 9\), and \(f(3) - f(-1) = 20\).

Step 1: Evaluate f(0)

Substitute \(x = 0\) into the function \(f(x) = 4x^2 - 3x + 2\). This gives \(f(0) = 4(0)^2 - 3(0) + 2 = 2\). Thus, \(f(0) = 2\).

Step 2: Evaluate f(3)

Substitute \(x = 3\) into the function. We calculate \(f(3) = 4(3)^2 - 3(3) + 2 = 4 \times 9 - 9 + 2 = 36 - 9 + 2 = 29\). So, \(f(3) = 29\).

Step 3: Evaluate f(-1)

Replace \(x = -1\) in the function. Compute \(f(-1) = 4(-1)^2 - 3(-1) + 2 = 4 \times 1 + 3 + 2 = 4 + 3 + 2 = 9\). Therefore, \(f(-1) = 9\).

Step 4: Calculate f(3) - f(-1)

To find \(f(3) - f(-1)\), subtract \(f(-1)\) from \(f(3)\). This gives \(f(3) - f(-1) = 29 - 9 = 20\).

Key Concepts

Function Evaluation

In mathematics, the evaluation of a polynomial function involves plugging specific values into the function to determine its result. When you encounter a problem that requires finding several evaluations, like evaluating a function for different values of \(x\), it's referred to as function evaluation. This process helps in understanding how a given function behaves with different inputs.
The polynomial function in this exercise is \(f(x)=4x^2-3x+2\). To evaluate a function, we simply replace the variable \(x\) with the desired value and follow through with the arithmetic operations:

• For \(f(0)\), substitute \(x = 0\), leading to \(f(0) = 4(0)^2 - 3(0) + 2 = 2\).
• For \(f(3)\), substitute \(x = 3\), leading to \(f(3) = 4(3)^2 - 3(3) + 2 = 29\).
• For \(f(-1)\), substitute \(x = -1\), leading to \(f(-1) = 4(-1)^2 - 3(-1) + 2 = 9\).

Each evaluation ultimately helps to find specific outputs at certain inputs. These specific points can give insight into the pattern or behavior of the function.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations like addition or multiplication. In the provided exercise, the expression \(4x^2 - 3x + 2\) represents a polynomial function of \(x\). Each component of this expression has a distinct role:

• The term \(4x^2\) is called a quadratic term, since the variable \(x\) is raised to the power of 2.
• \(-3x\) is referred to as the linear term due to the presence of the variable \(x\) to the power of 1.
• Lastly, \(+2\) represents the constant term, which stays the same regardless of the value of \(x\).

Algebraic expressions, specifically polynomials, are used to model various real-world situations. They provide a convenient and standardized way to represent relationships where one quantity depends on another.

Substitution Method

The substitution method, also known as plugging in values, is a fundamental technique in algebra used to evaluate expressions or solve equations. It's simply the action of replacing a variable with a numerical value or expression. This method is invaluable in simplifying and calculating the result of an expression.
Consider our polynomial \(f(x) = 4x^2 - 3x + 2\). To evaluate \(f(x)\) for specific values, we employ substitution by replacing \(x\) with numbers:

• Place \(x = 0\) in \(f(x)\) to find \(f(0) = 2\).
• Insert \(x = 3\) to compute \(f(3) = 29\).
• Replace \(x = -1\) to get \(f(-1) = 9\).

The substitution method is not limited to evaluating functions but also plays a crucial role in solving equations. In exercises that involve multiple values like ours, it becomes especially powerful, allowing for quick calculations and deeper insights into functional behavior.