Function Operations Explained Solving F(x) = 3x - 1 And G(x) = 9x² - 1

by ADMIN 71 views
Iklan Headers

In the realm of mathematics, function operations serve as fundamental tools for manipulating and combining functions to create new ones. This exploration delves into the intricate world of function operations, using the specific functions f(x) = 3x - 1 and g(x) = 9x² - 1 as our guiding examples. We will embark on a journey to unravel the mysteries behind addition, subtraction, multiplication, and division of functions, providing a comprehensive understanding of these core concepts.

a) (f + g)(x): The Sum of Functions

At the heart of function operations lies the concept of function addition. To determine the sum of two functions, denoted as (f + g)(x), we simply combine the expressions of the individual functions. In essence, we add the output of f(x) to the output of g(x) for every value of x within their shared domain. Let's unravel this concept further by applying it to our functions, f(x) = 3x - 1 and g(x) = 9x² - 1.

Understanding Function Addition

Function addition, represented as (f + g)(x), entails combining the outputs of two functions for a given input. It's akin to merging two streams of data, where each stream represents the output of a function. To calculate (f + g)(x), we simply add the expressions of f(x) and g(x) together. This operation yields a new function that embodies the combined behavior of the original two functions.

Applying Function Addition to f(x) and g(x)

To find (f + g)(x) for our functions f(x) = 3x - 1 and g(x) = 9x² - 1, we perform the following steps:

  1. Write down the expressions for f(x) and g(x):
    • f(x) = 3x - 1
    • g(x) = 9x² - 1
  2. Add the expressions together: (f + g)(x) = (3x - 1) + (9x² - 1)
  3. Simplify the expression by combining like terms: (f + g)(x) = 9x² + 3x - 2

Therefore, the sum of the functions f(x) and g(x) is (f + g)(x) = 9x² + 3x - 2. This new function represents the combined output of f(x) and g(x) for any given input x.

Delving Deeper into the Significance of Function Addition

Function addition is not merely a mathematical procedure; it holds significant practical implications. In various real-world scenarios, we often encounter situations where combining the effects of two functions provides valuable insights. For instance, consider a scenario where f(x) represents the cost of producing x units of a product, and g(x) represents the revenue generated from selling x units of the same product. In this case, (f + g)(x) would represent the total profit (or loss) for producing and selling x units.

Furthermore, function addition plays a crucial role in diverse fields such as physics, engineering, and economics. In physics, it can be used to combine forces or velocities. In engineering, it can be used to analyze the combined performance of different components in a system. In economics, it can be used to model the combined effects of different economic factors.

b) (f - g)(x): The Difference of Functions

In the realm of function operations, subtraction plays a pivotal role in understanding the relationship between two functions. The difference of two functions, denoted as (f - g)(x), reveals the disparity between their outputs for a given input. Essentially, we subtract the output of g(x) from the output of f(x) for each value of x within their common domain. Let's delve deeper into this concept by exploring its application to our functions, f(x) = 3x - 1 and g(x) = 9x² - 1.

Unveiling the Essence of Function Subtraction

Function subtraction, symbolized as (f - g)(x), involves determining the difference between the outputs of two functions for a specific input. It's akin to comparing two streams of data, where each stream represents the output of a function. To compute (f - g)(x), we subtract the expression of g(x) from the expression of f(x). This operation results in a new function that signifies the difference in behavior between the original two functions.

Applying Function Subtraction to f(x) and g(x)

To find (f - g)(x) for our functions f(x) = 3x - 1 and g(x) = 9x² - 1, we follow these steps:

  1. Write down the expressions for f(x) and g(x):
    • f(x) = 3x - 1
    • g(x) = 9x² - 1
  2. Subtract the expression of g(x) from the expression of f(x): (f - g)(x) = (3x - 1) - (9x² - 1)
  3. Simplify the expression by distributing the negative sign and combining like terms: (f - g)(x) = 3x - 1 - 9x² + 1 (f - g)(x) = -9x² + 3x

Therefore, the difference between the functions f(x) and g(x) is (f - g)(x) = -9x² + 3x. This new function represents the disparity in output between f(x) and g(x) for any given input x.

Exploring the Significance of Function Subtraction

Function subtraction transcends the realm of mere mathematical calculation; it holds profound practical implications. In numerous real-world scenarios, we encounter situations where discerning the difference between the effects of two functions provides invaluable insights. Consider, for example, a scenario where f(x) represents the revenue generated from selling x units of a product, and g(x) represents the cost of producing x units of the same product. In this context, (f - g)(x) would represent the profit earned from producing and selling x units.

Moreover, function subtraction plays a vital role in diverse fields such as physics, engineering, and economics. In physics, it can be used to determine the net force acting on an object. In engineering, it can be used to analyze the difference in performance between two designs. In economics, it can be used to model the impact of different economic policies.

c) (g * f)(x): The Product of Functions

In the realm of function operations, multiplication takes center stage as a powerful tool for combining functions in a unique way. The product of two functions, denoted as (g * f)(x), signifies the result of multiplying the outputs of the individual functions for a given input. In essence, we multiply the output of g(x) by the output of f(x) for each value of x within their shared domain. Let's embark on a journey to unravel this concept further by applying it to our functions, f(x) = 3x - 1 and g(x) = 9x² - 1.

Understanding the Essence of Function Multiplication

Function multiplication, represented as (g * f)(x), involves multiplying the outputs of two functions for a given input. It's akin to combining two streams of data in a multiplicative manner, where each stream represents the output of a function. To calculate (g * f)(x), we simply multiply the expressions of g(x) and f(x) together. This operation yields a new function that embodies the combined effect of the original two functions in a multiplicative sense.

Applying Function Multiplication to f(x) and g(x)

To find (g * f)(x) for our functions f(x) = 3x - 1 and g(x) = 9x² - 1, we perform the following steps:

  1. Write down the expressions for f(x) and g(x):
    • f(x) = 3x - 1
    • g(x) = 9x² - 1
  2. Multiply the expressions together: (g * f)(x) = (9x² - 1) * (3x - 1)
  3. Expand the expression by using the distributive property (or the FOIL method): (g * f)(x) = 9x² * (3x - 1) - 1 * (3x - 1) (g * f)(x) = 27x³ - 9x² - 3x + 1

Therefore, the product of the functions f(x) and g(x) is (g * f)(x) = 27x³ - 9x² - 3x + 1. This new function represents the combined output of g(x) and f(x) in a multiplicative manner for any given input x.

Delving Deeper into the Significance of Function Multiplication

Function multiplication transcends the realm of mere mathematical procedure; it holds significant practical implications. In various real-world scenarios, we often encounter situations where combining the effects of two functions in a multiplicative manner provides valuable insights. For instance, consider a scenario where f(x) represents the number of items sold at a certain price, and g(x) represents the price per item. In this case, (g * f)(x) would represent the total revenue generated from selling x items.

Furthermore, function multiplication plays a crucial role in diverse fields such as physics, engineering, and economics. In physics, it can be used to calculate work done by a force. In engineering, it can be used to analyze the power output of a system. In economics, it can be used to model the growth of an investment.

d) (f / g)(x): The Quotient of Functions

In the realm of function operations, division emerges as a crucial tool for comparing and relating functions. The quotient of two functions, denoted as (f / g)(x), represents the result of dividing the output of one function by the output of another function for a given input. In essence, we divide the output of f(x) by the output of g(x) for each value of x within their shared domain, with the crucial caveat that g(x) cannot be zero. Let's embark on a journey to unravel this concept further by applying it to our functions, f(x) = 3x - 1 and g(x) = 9x² - 1.

Understanding the Essence of Function Division

Function division, represented as (f / g)(x), involves dividing the output of one function by the output of another function for a specific input. It's akin to comparing two streams of data in a ratio-like manner, where each stream represents the output of a function. To calculate (f / g)(x), we simply divide the expression of f(x) by the expression of g(x). This operation yields a new function that embodies the relative behavior of the original two functions.

Applying Function Division to f(x) and g(x)

To find (f / g)(x) for our functions f(x) = 3x - 1 and g(x) = 9x² - 1, we perform the following steps:

  1. Write down the expressions for f(x) and g(x):
    • f(x) = 3x - 1
    • g(x) = 9x² - 1
  2. Divide the expression of f(x) by the expression of g(x): (f / g)(x) = (3x - 1) / (9x² - 1)
  3. Simplify the expression by factoring the denominator and canceling common factors, if possible: (f / g)(x) = (3x - 1) / [(3x - 1)(3x + 1)] (f / g)(x) = 1 / (3x + 1), where x ≠ 1/3 and x ≠ -1/3

Therefore, the quotient of the functions f(x) and g(x) is (f / g)(x) = 1 / (3x + 1), with the crucial restriction that x cannot be equal to 1/3 or -1/3, as these values would make the denominator zero, leading to an undefined result. This new function represents the relative output of f(x) compared to g(x) for any given input x, excluding the values that make the denominator zero.

Delving Deeper into the Significance of Function Division

Function division transcends the realm of mere mathematical procedure; it holds significant practical implications. In various real-world scenarios, we often encounter situations where comparing the effects of two functions in a ratio-like manner provides valuable insights. For instance, consider a scenario where f(x) represents the distance traveled by a car, and g(x) represents the time taken to travel that distance. In this case, (f / g)(x) would represent the average speed of the car.

Furthermore, function division plays a crucial role in diverse fields such as physics, engineering, and economics. In physics, it can be used to calculate density. In engineering, it can be used to analyze the efficiency of a system. In economics, it can be used to model price elasticity of demand.

In conclusion, function operations serve as indispensable tools in the world of mathematics and beyond. Through the exploration of addition, subtraction, multiplication, and division, we have gained a deeper understanding of how functions can be combined and manipulated to create new ones. The examples of f(x) = 3x - 1 and g(x) = 9x² - 1 have served as a practical guide, illuminating the power and versatility of function operations in solving mathematical problems and modeling real-world phenomena.