Function Operations Addition And Subtraction

Hey guys! Today, we're diving into the fascinating world of function operations. Think of functions like little machines that take an input, do some magic, and spit out an output. Now, what happens when we start combining these machines? That's where function operations come in! We're going to explore how to add, subtract, and generally mess around with functions to create new ones. So, buckle up, and let's get started!

1. Subtracting Functions: (h - g)(x)

Let's kick things off with our first example: h(x) = -8x + 6 and g(x) = 2x + 2. Our mission, should we choose to accept it, is to find (h - g)(x). What does this mean? Simply put, we need to subtract the function g(x) from the function h(x). Think of it like this: we're taking the output of g(x) away from the output of h(x) for the same input x.

The first crucial step in tackling function subtraction is setting up the expression correctly. We write (h - g)(x) = h(x) - g(x). This notation tells us exactly what operation to perform. Now, we substitute the actual function definitions: (h - g)(x) = (-8x + 6) - (2x + 2). Pay close attention to the parentheses here! They are super important because we are subtracting the entire function g(x), not just the first term.

Next up is distribution, often a critical step in algebraic manipulations. The minus sign in front of the parentheses acts like a -1 that we need to distribute to each term inside the parentheses of g(x). This means we multiply -1 by both 2x and +2. So, we get: (h - g)(x) = -8x + 6 - 2x - 2. See how the signs of the terms inside the second set of parentheses have changed? This is why those parentheses are so vital!

Now comes the satisfying part: combining like terms. We have two terms with 'x' (-8x and -2x) and two constant terms (+6 and -2). Let's group them together: (h - g)(x) = (-8x - 2x) + (6 - 2). Adding the 'x' terms, -8x - 2x gives us -10x. Combining the constants, 6 - 2 gives us 4. Putting it all together, we get our final answer: (h - g)(x) = -10x + 4.

So, what does this resulting function, -10x + 4, represent? It's a brand-new function that gives us the difference between the outputs of h(x) and g(x) for any given x. In essence, we've created a new machine that performs a combined operation based on our original two functions. Remember, the key to function subtraction lies in careful distribution and combining like terms. Get those steps right, and you'll be subtracting functions like a pro!

2. Adding Functions: (h + g)(x)

Alright, let's switch gears and talk about adding functions. In this scenario, we're given two new functions: h(x) = 5x + 3 and g(x) = 7x + 2. Our mission, should we choose to accept it (again!), is to find (h + g)(x). Just like subtraction, this notation tells us exactly what to do: add the function g(x) to the function h(x). Essentially, we're combining the outputs of these two functions for the same input x.

The first step, just like before, is setting up the expression correctly. We write (h + g)(x) = h(x) + g(x). This clearly shows us that we need to add the two functions together. Now, we substitute in the actual function definitions: (h + g)(x) = (5x + 3) + (7x + 2). Notice that we still use parentheses here, although in this case, they are less critical than in subtraction because there's no distribution of a negative sign to worry about. However, using parentheses consistently helps maintain clarity and avoid errors, especially when dealing with more complex functions.

Next, we focus on simplifying the expression. Since we are adding, we can simply remove the parentheses without changing any signs. This gives us: (h + g)(x) = 5x + 3 + 7x + 2. Now, it's time to combine those like terms again. We have two terms with 'x' (5x and 7x) and two constant terms (+3 and +2). Let's group them together: (h + g)(x) = (5x + 7x) + (3 + 2).

Adding the 'x' terms, 5x + 7x gives us 12x. Combining the constants, 3 + 2 gives us 5. Putting it all together, we get our final answer: (h + g)(x) = 12x + 5. Voila! We've successfully added two functions together.

So, what does this resulting function, 12x + 5, represent? It’s a new function that gives us the sum of the outputs of h(x) and g(x) for any given x. We’ve created a combined function that represents the total effect of both original functions. The beauty of function addition lies in its straightforward nature – simply combine like terms after adding the function expressions. With practice, you'll be adding functions in your sleep!

3. Combining Functions: (f + g)(x) with Polynomials

Now, let's crank up the complexity a little bit! This time, we're working with functions that involve higher powers of 'x', specifically polynomials. We have f(x) = 5x - 7 and g(x) = 4x^2 + 2x + 5. Our mission, as you might have guessed, is to find (f + g)(x). This means we're adding these two polynomial functions together. Don't let the squares and extra terms intimidate you; the process is still the same at its core.

The first step, as always, is to set up the expression correctly. We write (f + g)(x) = f(x) + g(x). This makes it clear that we need to add the function g(x) to the function f(x). Now, we substitute the actual function definitions: (f + g)(x) = (5x - 7) + (4x^2 + 2x + 5). Notice that g(x) is a quadratic function (it has an x^2 term), while f(x) is a linear function (highest power of x is 1). This means our resulting function might be a quadratic as well.

Next, we simplify the expression. Since we're adding, we can remove the parentheses without changing any signs: (f + g)(x) = 5x - 7 + 4x^2 + 2x + 5. Now comes the crucial step of combining like terms. This time, we have terms with x^2, terms with x, and constant terms. To keep things organized, it's often helpful to write the terms in descending order of their exponents. This is called standard form for polynomials.

So, let's rearrange the terms: (f + g)(x) = 4x^2 + 5x + 2x - 7 + 5. Now, we can easily identify and combine like terms. We only have one x^2 term, which is 4x^2. We have two 'x' terms: 5x and 2x. And we have two constant terms: -7 and +5.

Combining the 'x' terms, 5x + 2x gives us 7x. Combining the constants, -7 + 5 gives us -2. Putting it all together, we get our final answer: (f + g)(x) = 4x^2 + 7x - 2.

So, what does this resulting function, 4x^2 + 7x - 2, represent? It's a new quadratic function that gives us the sum of the outputs of f(x) and g(x) for any given x. We've successfully combined a linear and a quadratic function to create a new quadratic function. The key takeaway here is that when adding functions, especially polynomials, organization is crucial. Writing terms in descending order of exponents and carefully combining like terms will help you avoid errors and arrive at the correct result.

4. Subtracting Functions: (f - g)(x) with Polynomials

Let's dive back into the world of polynomial functions, but this time, we're going to subtract! We have f(x) = 6x^2 - 5x and g(x) = 3x^2 - 6x + 4. Our mission, should we choose to accept it (for the third time!), is to find (f - g)(x). This means we're subtracting the entire function g(x) from the function f(x). Remember those parentheses? They're about to become our best friends.

The first step, as always, is to set up the expression correctly. We write (f - g)(x) = f(x) - g(x). This makes it clear that we need to subtract g(x) from f(x). Now, we substitute the actual function definitions: (f - g)(x) = (6x^2 - 5x) - (3x^2 - 6x + 4). Notice the crucial use of parentheses around the entire expression for g(x). This is absolutely essential because we need to subtract every term in g(x).

Now comes the most important part: distributing the negative sign. The minus sign in front of the parentheses acts like a -1 that we need to multiply by each term inside the parentheses of g(x). This means we multiply -1 by 3x^2, -6x, and +4. Doing so, we get: (f - g)(x) = 6x^2 - 5x - 3x^2 + 6x - 4. See how the signs of the terms inside the second set of parentheses have flipped? This is why distribution is so critical in function subtraction!

Next up is combining like terms. We have terms with x^2, terms with x, and constant terms. Let's group them together, keeping in mind the standard form for polynomials (descending order of exponents): (f - g)(x) = (6x^2 - 3x^2) + (-5x + 6x) - 4.

Combining the x^2 terms, 6x^2 - 3x^2 gives us 3x^2. Combining the 'x' terms, -5x + 6x gives us 1x, which we can simply write as x. The constant term is just -4. Putting it all together, we get our final answer: (f - g)(x) = 3x^2 + x - 4.

So, what does this resulting function, 3x^2 + x - 4, represent? It's a new quadratic function that gives us the difference between the outputs of f(x) and g(x) for any given x. We've successfully subtracted one polynomial function from another. The key to subtracting polynomials (and functions in general) is to remember the importance of distributing the negative sign. Get that right, and you're golden!

5. Function Operations: A Quick Recap

Okay, guys, we've covered a lot of ground here! Let's do a quick recap to make sure we've got the key concepts down pat. We've explored how to add and subtract functions, both linear and polynomial. The core idea behind function operations is that we're combining the outputs of functions for the same input. Think of it like this: we're taking two (or more) function machines and hooking them up in a way that creates a new function machine.

Here are the key takeaways:

  • Notation Matters: The notation (h + g)(x), (h - g)(x), etc., tells us exactly what operation to perform.
  • Substitution is Key: The first step is always to substitute the actual function definitions into the expression.
  • Parentheses are Your Friends: Use parentheses, especially when subtracting, to ensure you distribute the negative sign correctly.
  • Combine Like Terms: After substituting and distributing (if necessary), combine like terms to simplify the expression.
  • Polynomials Need Order: When working with polynomials, write your final answer in standard form (descending order of exponents).

With these principles in mind, you'll be well-equipped to tackle all sorts of function operations. Remember, practice makes perfect! The more you work with these concepts, the more natural they will become. Keep up the great work, and you'll be a function operation master in no time!

Now that we've covered the theory, let's solidify your understanding with some practice problems. Working through these exercises will help you internalize the steps and build confidence in your abilities. Remember, the key to mastering any mathematical concept is consistent practice.

Problem 1:

Given p(x) = 3x^2 + 2x - 1 and q(x) = x^2 - 4x + 3, find (p + q)(x).

Problem 2:

Given r(x) = -2x + 5 and s(x) = 4x - 7, find (r - s)(x).

Problem 3:

Given a(x) = x^3 - 2x^2 + x and b(x) = -x^2 + 3x - 2, find (a + b)(x).

Problem 4:

Given c(x) = 5x^2 - 3x + 2 and d(x) = 2x^2 + x - 4, find (c - d)(x).

Problem 5:

Given m(x) = 7x - 1 and n(x) = -3x + 5, find (m + n)(x) and (m - n)(x).

Take your time to work through these problems, and don't hesitate to review the examples and explanations we discussed earlier. Remember to pay close attention to parentheses, distribute negative signs carefully, and combine like terms accurately. Good luck, and happy problem-solving!

Alright, let's check your work and see how you did on those practice problems. It's important to not only get the right answers but also to understand the process behind them. So, even if you made a mistake, don't worry! We're going to walk through each solution step-by-step.

Solution to Problem 1:

Given p(x) = 3x^2 + 2x - 1 and q(x) = x^2 - 4x + 3, find (p + q)(x).

  1. Set up the expression: (p + q)(x) = p(x) + q(x)
  2. Substitute the function definitions: (p + q)(x) = (3x^2 + 2x - 1) + (x^2 - 4x + 3)
  3. Remove parentheses: (p + q)(x) = 3x^2 + 2x - 1 + x^2 - 4x + 3
  4. Combine like terms: (p + q)(x) = (3x^2 + x^2) + (2x - 4x) + (-1 + 3)
  5. Simplify: (p + q)(x) = 4x^2 - 2x + 2

So, the solution to Problem 1 is (p + q)(x) = 4x^2 - 2x + 2.

Solution to Problem 2:

Given r(x) = -2x + 5 and s(x) = 4x - 7, find (r - s)(x).

  1. Set up the expression: (r - s)(x) = r(x) - s(x)
  2. Substitute the function definitions: (r - s)(x) = (-2x + 5) - (4x - 7)
  3. Distribute the negative sign: (r - s)(x) = -2x + 5 - 4x + 7
  4. Combine like terms: (r - s)(x) = (-2x - 4x) + (5 + 7)
  5. Simplify: (r - s)(x) = -6x + 12

So, the solution to Problem 2 is (r - s)(x) = -6x + 12.

Solution to Problem 3:

Given a(x) = x^3 - 2x^2 + x and b(x) = -x^2 + 3x - 2, find (a + b)(x).

  1. Set up the expression: (a + b)(x) = a(x) + b(x)
  2. Substitute the function definitions: (a + b)(x) = (x^3 - 2x^2 + x) + (-x^2 + 3x - 2)
  3. Remove parentheses: (a + b)(x) = x^3 - 2x^2 + x - x^2 + 3x - 2
  4. Combine like terms: (a + b)(x) = x^3 + (-2x^2 - x^2) + (x + 3x) - 2
  5. Simplify: (a + b)(x) = x^3 - 3x^2 + 4x - 2

So, the solution to Problem 3 is (a + b)(x) = x^3 - 3x^2 + 4x - 2.

Solution to Problem 4:

Given c(x) = 5x^2 - 3x + 2 and d(x) = 2x^2 + x - 4, find (c - d)(x).

  1. Set up the expression: (c - d)(x) = c(x) - d(x)
  2. Substitute the function definitions: (c - d)(x) = (5x^2 - 3x + 2) - (2x^2 + x - 4)
  3. Distribute the negative sign: (c - d)(x) = 5x^2 - 3x + 2 - 2x^2 - x + 4
  4. Combine like terms: (c - d)(x) = (5x^2 - 2x^2) + (-3x - x) + (2 + 4)
  5. Simplify: (c - d)(x) = 3x^2 - 4x + 6

So, the solution to Problem 4 is (c - d)(x) = 3x^2 - 4x + 6.

Solution to Problem 5:

Given m(x) = 7x - 1 and n(x) = -3x + 5, find (m + n)(x) and (m - n)(x).

  • Finding (m + n)(x):
    1. Set up the expression: (m + n)(x) = m(x) + n(x)
    2. Substitute the function definitions: (m + n)(x) = (7x - 1) + (-3x + 5)
    3. Remove parentheses: (m + n)(x) = 7x - 1 - 3x + 5
    4. Combine like terms: (m + n)(x) = (7x - 3x) + (-1 + 5)
    5. Simplify: (m + n)(x) = 4x + 4
  • Finding (m - n)(x):
    1. Set up the expression: (m - n)(x) = m(x) - n(x)
    2. Substitute the function definitions: (m - n)(x) = (7x - 1) - (-3x + 5)
    3. Distribute the negative sign: (m - n)(x) = 7x - 1 + 3x - 5
    4. Combine like terms: (m - n)(x) = (7x + 3x) + (-1 - 5)
    5. Simplify: (m - n)(x) = 10x - 6

So, the solutions to Problem 5 are (m + n)(x) = 4x + 4 and (m - n)(x) = 10x - 6.

How did you do? If you got all the answers right, congratulations! You've clearly mastered the concepts of adding and subtracting functions. If you made a few mistakes, don't worry! Review the steps where you struggled, and try working through the problems again. Remember, practice is key to success. Keep up the great work, and you'll be a function operation expert in no time!