Question

Suppose the function \(f\) expects two numeric values as its inputs and returns their addition as its output value, and \(g\) is a function that returns the subtraction of the two values given as its input. If \(a\) and \(b\) represent numeric values, what is the result returned by \(f(f(a, b), g(a, b))\) ?

Short answer

The result is \(2a\).

Step-by-step solution

Understanding Function f

The function \(f(x, y)\) takes two inputs, \(x\) and \(y\), and returns their sum. Therefore, \(f(a, b) = a + b\).

Understanding Function g

The function \(g(x, y)\) takes two inputs, \(x\) and \(y\), and returns their difference. Therefore, \(g(a, b) = a - b\).

Substituting into the Composite Function

To find \(f(f(a, b), g(a, b))\), we need to evaluate the inner functions first and then replace their results in the outer function. We have \(f(a, b) = a + b\) and \(g(a, b) = a - b\).

Evaluating the Composite Function

Now calculate \(f(f(a, b), g(a, b))\) by substituting the previous results: \(f(a + b, a - b)\). Since \(f(x, y) = x + y\), replacing \(x\) with \(a + b\) and \(y\) with \(a - b\), we get \((a + b) + (a - b)\).

Simplifying the Expression

Simplify the expression \((a + b) + (a - b)\): The terms \(b\) and \(-b\) cancel each other out, leaving us with \(a + a\), which simplifies to \(2a\).

Key concepts

Addition Function

An addition function is a simple yet fundamental concept in mathematics. It involves taking two numbers, often called operands, and calculating their sum. This operation can be represented by the function \( f(x, y) = x + y \). In this exercise, the function \( f \) is used to add two numbers \( a \) and \( b \).

When you plug in values to this addition function, it returns the total or the sum. For example, if \( f(3, 5) \), it will compute \( 3 + 5 \) giving a result of 8. The idea of adding numbers is a building block for more complex mathematical concepts.

It's essential to understand addition functions because they are widely used in various equations and real-world problem solving.

Subtraction Function

A subtraction function is another basic, yet crucial mathematical operation. It determines the difference between two numbers. Mathematically, it is expressed as \( g(x, y) = x - y \). In our given exercise, this function takes the numbers \( a \) and \( b \) and calculates their subtraction.

The subtraction operation helps you find out how much one number differs from another. For instance, using our subtraction function \( g(10, 4) \), calculates \( 10 - 4 \), resulting in 6.

Understanding subtraction functions forms the groundwork for comprehending other compound functions and problem-solving scenarios.

Composite Functions

Composite functions involve combining two or more functions to create a new function. This is often done to simplify complex calculations or to conduct operations in a specific sequence. In the current problem, a composite function \( f(f(a, b), g(a, b)) \) is explored.

The operation starts by solving the inner functions first. You first evaluate \( f(a, b) = a + b \) and \( g(a, b) = a - b \). These results are then used as arguments for the outer function \( f \). This results in the evaluation of \( f(f(a, b), g(a, b)) = f(a + b, a - b) \), which simplifies to \( (a + b) + (a - b) \).

Finally, the expression simplifies to \( 2a \), illustrating how composite functions can combine multiple operations into a singular, more manageable expression.

Mathematical Operations

Mathematical operations such as addition and subtraction form the core tools for manipulating numbers. They allow you to express and solve equations effectively. These operations form the basis of broader mathematical concepts, from basic arithmetic to complex calculus.

In the given exercise, understanding the individual operations \( + \) and \( - \) enables you to solve the composite function \( f(f(a, b), g(a, b)) \). After carrying out the operations, simplifying \( (a + b) + (a - b) \) ultimately leads to the solution \( 2a \).

Grasping these fundamental operations is crucial as they not only solve everyday problems but also create a foundation for advanced mathematical studies and real-life applications. By mastering these basic operations, you gain the ability to approach more complex problems with confidence.