Problem 1 - Calculus: One and Several Variables

Question

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

Step-by-Step Solution

Answer

The solutions for the function \(f(x)\) at the four points are \(f(0) = 2, f(1) = 1, f(-2) = 16\) and \(f(3/2) = 2.\)

Step 1: Substitute \(x = 0\) in the function

Firstly, substitute \(x = 0\) into the function to get \(f(0)\). This gives \(f(0) = 2*(0)^2 - 3*0 + 2 = 0 - 0 + 2 = 2.\)

Step 2: Substitute \(x = 1\) in the function

Next, let \(x = 1\) and substitute it into the function to calculate \(f(1)\). This results in \(f(1) = 2*(1)^2 - 3*1 + 2 = 2 - 3 + 2 = 1.

Step 3: Substitute \(x = -2\) in the function

Then, for \(x = -2\), the calculation of \(f(-2)\) is done by substituting into the function. When \(x = -2\), it results in \(f(-2) = 2*(-2)^2 - 3*(-2) + 2 = 8 + 6 + 2 = 16.\)

Step 4: Substitute \(x = 3/2\) in the function

Lastly, by substituting \(x = 3/2\) into the function, \(f(3/2)\) can be found. This leads to \(f(3/2) = 2*(3/2)^2 - 3*(3/2) + 2 = 9/2 - 9/2 + 2 = 2.\)

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in mathematics, especially when dealing with expressions like quadratic functions. It involves finding the value of a function for specific input values. For a given quadratic function like \( f(x) = 2x^2 - 3x + 2 \), function evaluation lets you determine outputs for different inputs.
To evaluate, you substitute the given \( x \) values into the function:

• When \( x = 0 \), substituting gives \( f(0) = 2(0)^2 - 3(0) + 2 = 2 \).
• For \( x = 1 \), it becomes \( f(1) = 2(1)^2 - 3(1) + 2 = 1 \).
• With \( x = -2 \), the result is \( f(-2) = 16 \).
• Finally, for \( x = \frac{3}{2} \), we find \( f\left(\frac{3}{2}\right) = 2 \).

Each evaluation provides important insights into the behavior of the function for specific values and helps in graphing or solving more complex problems.

Substitution Method

The substitution method is a straightforward technique used to evaluate expressions. This method involves replacing a variable with a known value to simplify and solve the expression.
In the context of quadratic functions like \( f(x) = 2x^2 - 3x + 2 \), the substitution method helps find specific function values:

• To find \( f(0) \), substitute \( x = 0 \) into the function: \( 2(0)^2 - 3(0) + 2 = 2 \).
• For \( f(1) \), use \( x = 1 \) to get \( 2(1)^2 - 3(1) + 2 = 1 \).
• Substituting \( x = -2 \) reveals \( f(-2) = 16 \).
• Lastly, replacing \( x \) with \( \frac{3}{2} \) gives \( f\left(\frac{3}{2}\right) = 2 \).

This technique simplifies complex expressions and allows for easier computation, especially within polynomial functions.

Polynomial Functions

Polynomial functions are expressions involving variables and coefficients, with operations of addition, subtraction, and non-negative integer exponents. Quadratic functions such as \( f(x) = 2x^2 - 3x + 2 \) are second-degree polynomial functions and are characterized by their parabolic graphs.
Key aspects of polynomial functions include:

• Degree: Defined by the highest exponent of the variable, the degree in this case is 2.
• Coefficients: These are the numerical factors of the terms. Here, 2 is associated with \( x^2 \), while -3 and 2 are the coefficients of \( x \) and the constant term, respectively.
• Graph: The graph of a quadratic function is a parabola that can open upwards or downwards depending on the sign of the leading coefficient.

Understanding polynomial functions and their characteristics aids in solving various algebraic problems, predicting function behavior, and analyzing real-world scenarios.