Finding (s+r)(x) A Comprehensive Guide To Function Addition

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In the realm of mathematics, functions play a pivotal role, serving as the building blocks for more complex mathematical structures. Among the various operations that can be performed on functions, addition stands out as a fundamental operation, allowing us to combine the outputs of two or more functions. In this comprehensive guide, we delve into the intricacies of adding functions, specifically focusing on the sum of two functions, denoted as (s+r)(x). We will explore the definition of function addition, dissect the notation (s+r)(x), and illustrate the process with a concrete example, providing a step-by-step solution to determine the expression for (s+r)(x) given the functions r(x) = 5x³ and s(x) = x².

Understanding Function Addition

Function addition is a straightforward operation that combines the outputs of two functions for a given input. To grasp the concept, let's consider two functions, f(x) and g(x). The sum of these functions, denoted as (f+g)(x), is defined as the sum of their individual outputs for the same input x. Mathematically, this can be expressed as:

(f+g)(x) = f(x) + g(x)

This equation elucidates that to find the value of the sum function (f+g)(x) at a specific input x, we simply add the values of the individual functions f(x) and g(x) at that same input.

Deciphering the Notation (s+r)(x)

The notation (s+r)(x) represents the sum of two specific functions, s(x) and r(x). Here, 's' and 'r' are the names of the functions, and 'x' is the input variable. The notation (s+r)(x) signifies that we are adding the outputs of the functions s(x) and r(x) for a given input x. In essence, (s+r)(x) is equivalent to s(x) + r(x).

To further solidify your understanding, let's break down the components of the notation:

  • s(x): This represents the function named 's' with input variable 'x'. It denotes the output of the function 's' when the input is 'x'.
  • r(x): Similarly, this represents the function named 'r' with input variable 'x'. It denotes the output of the function 'r' when the input is 'x'.
  • (s+r)(x): This represents the sum of the functions s(x) and r(x). It signifies the output obtained by adding the outputs of s(x) and r(x) for the same input 'x'.

Example: Finding (s+r)(x) for r(x) = 5x³ and s(x) = x²

To illustrate the process of finding (s+r)(x), let's consider a concrete example. Suppose we are given two functions:

  • r(x) = 5x³
  • s(x) = x²

Our goal is to determine the expression for (s+r)(x). Following the definition of function addition, we know that (s+r)(x) = s(x) + r(x). Therefore, to find (s+r)(x), we simply need to add the expressions for s(x) and r(x).

Step-by-Step Solution

Let's walk through the steps to find (s+r)(x):

  1. Write down the expressions for s(x) and r(x):

    • s(x) = x²
    • r(x) = 5x³

  2. Apply the definition of function addition:

    • (s+r)(x) = s(x) + r(x)

  3. Substitute the expressions for s(x) and r(x) into the equation:

    • (s+r)(x) = x² + 5x³

Therefore, the expression for (s+r)(x) is x² + 5x³.

Arranging the Terms (Optional)

While the expression x² + 5x³ is a correct representation of (s+r)(x), it is often customary to arrange the terms in descending order of their exponents. This means writing the term with the highest exponent first, followed by the term with the next highest exponent, and so on. Applying this convention, we can rewrite the expression as:

(s+r)(x) = 5x³ + x²

This expression is mathematically equivalent to x² + 5x³, but it is presented in a more conventional and organized manner.

Key Takeaways

  • Function addition is a fundamental operation that combines the outputs of two functions for a given input.
  • The notation (s+r)(x) represents the sum of the functions s(x) and r(x), which is equivalent to s(x) + r(x).
  • To find (s+r)(x), simply add the expressions for s(x) and r(x).
  • The terms in the expression for (s+r)(x) can be arranged in descending order of their exponents for better readability.

Additional Examples and Practice Problems

To further enhance your understanding of function addition, let's explore some additional examples and practice problems.

Example 1:

Suppose f(x) = 2x + 1 and g(x) = x² - 3. Find (f+g)(x).

Solution:

(f+g)(x) = f(x) + g(x) = (2x + 1) + (x² - 3) = x² + 2x - 2

Example 2:

Suppose h(x) = √x and k(x) = 4x - 2. Find (h+k)(x).

Solution:

(h+k)(x) = h(x) + k(x) = √x + (4x - 2) = 4x + √x - 2

Practice Problems:

  1. Let p(x) = 3x² - x + 5 and q(x) = -2x² + 4x - 1. Find (p+q)(x).
  2. Let m(x) = |x| and n(x) = x³. Find (m+n)(x).
  3. Let a(x) = 1/(x+1) and b(x) = x/(x-1). Find (a+b)(x).

Applications of Function Addition

Function addition is not merely a theoretical concept; it has practical applications in various fields, including:

  • Physics: In physics, function addition can be used to combine the effects of multiple forces acting on an object. For example, if two forces, F1(t) and F2(t), are acting on an object at time t, the net force acting on the object is given by (F1+F2)(t).
  • Economics: In economics, function addition can be used to model the total cost or revenue of a company. For example, if C(x) represents the cost of producing x units of a product and R(x) represents the revenue generated from selling x units, the total profit is given by (R-C)(x), which involves function addition and subtraction.
  • Computer Graphics: In computer graphics, function addition can be used to combine different transformations applied to an object. For example, if T1(x,y) and T2(x,y) represent two transformations applied to a point (x,y), the combined transformation is given by (T1+T2)(x,y).

Conclusion

Function addition is a fundamental operation in mathematics that allows us to combine the outputs of two or more functions. The notation (s+r)(x) represents the sum of the functions s(x) and r(x), and it is equivalent to s(x) + r(x). To find (s+r)(x), we simply add the expressions for s(x) and r(x). Function addition has practical applications in various fields, including physics, economics, and computer graphics. By mastering the concept of function addition, you will gain a deeper understanding of functions and their role in mathematics and beyond.

This comprehensive guide has provided you with a thorough understanding of function addition, specifically focusing on the sum of two functions, (s+r)(x). You have learned the definition of function addition, dissected the notation (s+r)(x), and worked through a concrete example to determine the expression for (s+r)(x) given the functions r(x) = 5x³ and s(x) = x². With the knowledge and practice problems provided, you are well-equipped to tackle more complex problems involving function addition.

Remember, the key to success in mathematics lies in consistent practice and a solid understanding of the fundamental concepts. So, continue to explore the world of functions, delve deeper into their properties and operations, and you will undoubtedly excel in your mathematical journey.