Problem 48 - Calculus with Concepts in Calculus

Question

Find a formula for the function \(f\) that assigns to each \(x\) greater than \(-1\) the number obtained by squaring \(x\), then subtracting \(2 x\), and finally adding \(\sqrt{2}\).

Step-by-Step Solution

Answer

The function is defined as \(f(x) = x^2 - 2x + \sqrt{2}\).

Step 1: Identify the Operations

The exercise requires three operations: squaring the variable \(x\), subtracting \(2x\), and adding \(\sqrt{2}\). We need to combine these operations to derive the formula.

Step 2: Square the Variable

Begin by squaring the variable \(x\). This results in the term \(x^2\). This is the first part of our function.

Step 3: Subtract Twice the Variable

Next, reduce the result by \(2x\). This means we must subtract \(2x\) from our squared term. Our expression now is \(x^2 - 2x\).

Step 4: Add Constant Term

Finally, we add \(\sqrt{2}\) to the expression from the previous step. This gives the final expression: \(x^2 - 2x + \sqrt{2}\).

Step 5: Write the Function

Combine all the steps into a single function expression. The function \(f(x)\) is defined as \(f(x) = x^2 - 2x + \sqrt{2}\).

Key Concepts

Operations in Functions

Understanding operations in functions is vital when working with mathematical expressions and equations. Function operations involve applying specific mathematical actions sequentially to transform an input, which is usually represented by a variable like \(x\). In our exercise, each operation plays a crucial role:

• Squaring \(x\).
• Subtracting twice the value of \(x\).
• Adding the constant \(\sqrt{2}\).

These operations must be executed in a specific order to ensure the result is accurate. It's like following a recipe; skipping or rearranging a step can change the final outcome. By carefully combining these operations, you transform \(x\) into its final expression \( f(x) = x^2 - 2x + \sqrt{2} \). This ordered sequence forms the backbone of many mathematical functions.

Squaring Operations

Squaring a number means multiplying it by itself. It's one of the fundamental operations in algebra. When you square a variable like \(x\), it becomes \(x^2\). Squaring changes the dimension of \(x\) from linear to squared, which can drastically increase its value, especially for larger numbers.

In our function, squaring \(x\) is the first operation, setting the stage for the entire expression. It's important to remember that squaring a negative number results in a positive outcome, since negative times negative yields a positive. This makes it essential to know that the squaring operation alters the nature of any given number, and it must be handled with precision to preserve the integrity of the mathematical expression.

Constant Terms

A constant term in a function is a value that does not change, regardless of the input value \(x\). It acts as a fixed value that is either added or subtracted from the rest of the expression. For our function \(f(x) = x^2 - 2x + \sqrt{2}\), the constant term is \(\sqrt{2}\).

Adding a constant term shifts the graph of the function vertically without affecting its shape. No matter what the value of \(x\) is, the constant term remains the same. This property is useful for adjusting the position of the function's graph relative to the x-axis, which can be a powerful tool in both algebra and calculus for interpreting functions' behaviors and characteristics.