Problem 16 - Algebra 2 and Trigonometry

Question

Create your own function \(\mathrm{f}(x)\) and show that \(\mathrm{f}(x)=2 \mathrm{f}(x)\) . Explain why this result is true in general.

Step-by-Step Solution

Answer

If \( f(x) = 0 \), then \( f(x) = 2f(x) \) because both equal zero.

Step 1: Create a Function

Let's create a simple function \( f(x) \). For example, we can choose \( f(x) = 0 \). This is a valid function because it assigns a value to every input \( x \). Thus, \( f(x) = 0 \) for all \( x \).

Step 2: Calculate 2f(x)

Calculate \( 2f(x) \) for our chosen function. Since \( f(x) = 0 \), multiplying by 2 gives \( 2f(x) = 2 \times 0 = 0 \).

Step 3: Verify f(x) = 2f(x)

Now, we compare \( f(x) \) and \( 2f(x) \). Since \( f(x) = 0 \) and \( 2f(x) = 0 \), clearly \( f(x) = 2f(x) \).

Step 4: Generalization

The reason \( f(x) = 2f(x) \) holds in this case is because any zero function will satisfy this condition: multiplying zero by any constant still results in zero. Thus, when \( f(x) = 0 \), not just 2 but any multiplication maintains \( f(x) = cf(x) \) for any constant \( c \).

Key Concepts

Function Definition

Before diving into more complex concepts like functionality, it's essential to understand what a function is. In mathematics, a function is a relationship that assigns exactly one output for each input. This is sometimes referred to as a "rule" or "mapping" process. For example, you might have a function like \( f(x) = x^2 \), which tells you to square whatever the input \( x \) is. The key thing to remember is that every input gives you one and only one output, making the function "well-defined."

• The input is usually denoted by \( x \).
• The output is \( f(x) \).
• The rule of the function defines how the input is transformed to an output.

In the context of elementary functions, understanding simple cases like constant functions is beneficial. Constant functions output the same value, no matter the input. For instance, the zero function \( f(x) = 0 \) is a type of constant function because its output is always zero.

Multiplication of Functions

Multiplying functions involves taking two functions \( f(x) \) and \( g(x) \), and forming a new function \( (f \, g)(x) = f(x) \cdot g(x) \). When the functions are numbers or constants for any given input, it simplifies to simple arithmetic.Let's explore this with the zero function already defined: \( f(x) = 0 \). If you multiply the zero function by any function or constant, the result will still be zero:

• \( 2f(x) = 2 \cdot 0 = 0 \)
• \( cf(x) = c \cdot 0 = 0 \) for any constant \( c \)

This is because zero multiplied by any value stays zero. It shows that any scalar multiplication involving a zero function maintains zero. One of the neat properties of multiplication involving zero.

Constants in Functions

Constants in functions have specific characteristics that make them relatively easy to handle. When a function outputs the same number, regardless of the input, it's called a constant function. An excellent way to grasp this is by revisiting the zero function, \( f(x) = 0 \).

• Every input \( x \) gives the same output \( f(x) = 0 \).
• Since multiplying any number by zero gives zero, the value remains constant.
• When a constant function is multiplied by any scalar \( c \), it results in another constant function.

For example, if you have a constant value in a function such as \( f(x) = 3 \) and you multiply it by a constant \( 2 \), you will have \( 2 \cdot 3 = 6 \). Thus, \( g(x) = 6 \), showing another constant function. But with zero functions, multiplying doesn't change the value; it's like multiplying with 1 for non-zero numbers.