Solving Function Operations Finding (h+g)(10) For H(x) = 3x + 3 And G(x) = -4x + 1

Introduction

In the realm of mathematics, function operations are a fundamental concept that allows us to combine and manipulate functions in various ways. Among these operations, the addition of functions is a common and straightforward procedure. This article delves into the process of finding the sum of two functions and evaluating the result at a specific point. We will focus on the specific example of finding $(h+g)(10)$ given the functions $h(x) = 3x + 3$ and $g(x) = -4x + 1$. This exploration will not only demonstrate the mechanics of function addition but also highlight the importance of understanding function notation and evaluation. By the end of this article, you will have a solid grasp of how to add functions and apply the result to solve problems.

Defining Function Operations

Before we dive into the specific problem, it's crucial to define what we mean by function operations. Just like we can perform arithmetic operations (addition, subtraction, multiplication, division) on numbers, we can also perform these operations on functions. When we add two functions, say $h(x)$ and $g(x)$, we are essentially creating a new function, denoted as $(h+g)(x)$, whose value at any point $x$ is the sum of the values of $h(x)$ and $g(x)$ at that point. Mathematically, this is expressed as:

$$(h+g)(x) = h(x) + g(x)$$

This definition forms the basis for our approach to solving the problem at hand. We will first find the expression for the sum of the functions $h(x)$ and $g(x)$, and then we will evaluate this combined function at the specific point $x = 10$. This process involves substituting the given expressions for $h(x)$ and $g(x)$, simplifying the resulting expression, and finally, plugging in the value of $x$ to obtain the desired result.

Problem Statement: Finding (h+g)(10)

Our primary goal is to determine the value of $(h+g)(10)$, given the functions:

• $h(x) = 3x + 3$
• $g(x) = -4x + 1$

This problem requires us to first add the functions $h(x)$ and $g(x)$ together, creating a new function $(h+g)(x)$. Once we have this combined function, we will substitute $x = 10$ into the expression to find the value of $(h+g)(10)$. This step-by-step approach ensures a clear and logical solution.

Step-by-Step Solution

Step 1: Adding the Functions h(x) and g(x)

To find $(h+g)(x)$, we simply add the expressions for $h(x)$ and $g(x)$:

$$(h+g)(x) = h(x) + g(x)$$

Substitute the given expressions:

$$(h+g)(x) = (3x + 3) + (-4x + 1)$$

Now, we combine like terms. This involves grouping together the terms with $x$ and the constant terms:

$$(h+g)(x) = 3x - 4x + 3 + 1$$

Simplify the expression:

$$(h+g)(x) = -x + 4$$

So, the sum of the functions $h(x)$ and $g(x)$ is a new function $(h+g)(x) = -x + 4$. This result is crucial because it allows us to easily evaluate the combined function at any value of $x$.

Step 2: Evaluating (h+g)(x) at x = 10

Now that we have found the expression for $(h+g)(x)$, we can evaluate it at $x = 10$. This means substituting $x = 10$ into the expression we just derived:

$$(h+g)(10) = -(10) + 4$$

Perform the arithmetic:

$$(h+g)(10) = -10 + 4$$

$$(h+g)(10) = -6$$

Therefore, the value of $(h+g)(10)$ is -6. This is the final answer to our problem.

Detailed Explanation of Each Step

Adding Functions: A Closer Look

The process of adding functions might seem straightforward, but it's essential to understand the underlying principles. When we add $h(x)$ and $g(x)$, we are creating a new function that represents the sum of the outputs of the individual functions for a given input $x$. This new function, $(h+g)(x)$, inherits the domain of both $h(x)$ and $g(x)$, meaning it is defined for all values of $x$ for which both $h(x)$ and $g(x)$ are defined. In our case, both $h(x)$ and $g(x)$ are linear functions, which are defined for all real numbers, so $(h+g)(x)$ is also defined for all real numbers.

When adding the expressions for $h(x)$ and $g(x)$, we combined like terms. This is a fundamental algebraic technique that involves grouping together terms with the same variable and exponent. In our case, we grouped the terms with $x$ ($3x$ and $-4x$) and the constant terms ($3$ and $1$). This simplification process is crucial for obtaining the simplest form of the combined function, which makes it easier to evaluate.

Evaluating Functions: The Substitution Principle

Evaluating a function at a specific point is a core concept in mathematics. It involves substituting the given value of the variable (in our case, $x$) into the function's expression and simplifying. This process gives us the output of the function for that specific input. In our problem, we evaluated $(h+g)(x)$ at $x = 10$. This meant replacing every instance of $x$ in the expression $-x + 4$ with the number 10. The resulting expression, $-10 + 4$, is a simple arithmetic calculation that yields the value of the function at $x = 10$.

It's important to note that the order of operations (PEMDAS/BODMAS) must be followed when evaluating functions. This ensures that we perform the operations in the correct sequence, leading to the correct result. In our case, we first performed the substitution and then the addition, following the order of operations.

Alternative Approaches and Common Mistakes

Alternative Approach: Evaluating h(10) and g(10) Separately

While we solved the problem by first finding $(h+g)(x)$ and then evaluating it at $x = 10$, there's an alternative approach. We could have first evaluated $h(10)$ and $g(10)$ separately and then added the results. Let's see how this works:

• Evaluate $h(10)$:

  $$h(10) = 3(10) + 3 = 30 + 3 = 33$$

• Evaluate $g(10)$:

  $$g(10) = -4(10) + 1 = -40 + 1 = -39$$

• Add the results:

  $$h(10) + g(10) = 33 + (-39) = -6$$

As you can see, we arrive at the same answer, -6. This alternative approach reinforces the understanding that $(h+g)(10)$ is indeed the sum of $h(10)$ and $g(10)$.

Common Mistakes to Avoid

When working with function operations, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure accurate solutions.

• Incorrectly distributing the negative sign: When adding functions, especially when one of the functions has a negative sign, it's crucial to distribute the negative sign correctly. For example, if we were subtracting functions, say $(h-g)(x)$, we would need to distribute the negative sign to all terms in $g(x)$.
• Combining unlike terms: A common mistake is to combine terms that are not like terms. For example, adding $3x$ and $3$ as $6x$ is incorrect. Only terms with the same variable and exponent can be combined.
• Incorrectly substituting values: When evaluating a function, it's essential to substitute the value correctly. Make sure you replace every instance of the variable with the given value.
• Forgetting the order of operations: As mentioned earlier, the order of operations (PEMDAS/BODMAS) is crucial. Failing to follow the correct order can lead to incorrect results.

By being mindful of these common mistakes, you can improve your accuracy and confidence in solving function operation problems.

Real-World Applications of Function Operations

Function operations are not just abstract mathematical concepts; they have numerous real-world applications. Understanding how to combine functions can help us model and solve problems in various fields.

• Economics: In economics, we might use functions to represent cost, revenue, and profit. For example, if $C(x)$ represents the cost of producing $x$ units and $R(x)$ represents the revenue from selling $x$ units, then the profit function $P(x)$ can be expressed as $P(x) = R(x) - C(x)$. This is an example of function subtraction.
• Physics: In physics, we might use functions to describe the position, velocity, and acceleration of an object. For example, if $s(t)$ represents the position of an object at time $t$ and $v(t)$ represents its velocity at time $t$, then the average velocity over an interval $[t_1, t_2]$ can be calculated using function operations.
• Computer Science: In computer science, functions are used extensively in programming. Function operations can be used to combine different algorithms or processes. For example, we might have two functions, one that encrypts data and another that compresses data. We could combine these functions to create a new function that first encrypts and then compresses the data.
• Everyday Life: Even in everyday life, we encounter function operations without realizing it. For example, if you have a coupon that gives you a percentage discount and a fixed amount off, you can think of these as two functions. The total discount you receive is the result of combining these two functions.

These are just a few examples of how function operations are used in the real world. The ability to combine and manipulate functions is a powerful tool for problem-solving and modeling complex situations.

Conclusion

In this article, we have explored the concept of function operations, specifically the addition of functions. We successfully found $(h+g)(10)$ for the functions $h(x) = 3x + 3$ and $g(x) = -4x + 1$. We achieved this by first finding the expression for the combined function $(h+g)(x)$ and then evaluating it at $x = 10$. We also discussed an alternative approach where we evaluated $h(10)$ and $g(10)$ separately and then added the results. Additionally, we highlighted common mistakes to avoid and explored real-world applications of function operations.

Understanding function operations is crucial for success in mathematics and related fields. It allows us to combine and manipulate functions in meaningful ways, enabling us to model and solve a wide range of problems. By mastering these concepts, you will be well-equipped to tackle more advanced mathematical topics and real-world applications.