Find P For (f-g)(p) = 4 On [0, ∞)

Let's dive into this math problem together, guys! We're given two functions, f(x) = x² + 3x + 4 and g(x) = -5x + 84, both defined on the interval from 0 to infinity. Our mission, should we choose to accept it (and we do!), is to find the value of p within this interval such that (f-g)(p) = 4. Sounds like a fun little quest, right?

Understanding the Problem

First, let's break down what the problem is really asking. We have two functions, f(x) and g(x), and we need to find a value p that makes the difference between these functions equal to 4. It's like saying, "Hey, at what point does the gap between these two curves become exactly 4?" The interval [0, ∞) simply tells us where to look for this p – it has to be a non-negative number.

Keywords to keep in mind here are: functions, interval, value of p, (f-g)(p) = 4. These are the core concepts we'll be working with.

Setting up the Equation

To find this p, we first need to understand what (f-g)(p) means. It's simply the function f evaluated at p minus the function g evaluated at p. Mathematically, this is written as:

(f-g)(p) = f(p) - g(p)

Now, let's plug in the actual functions:

(f-g)(p) = (p² + 3p + 4) - (-5p + 84)

Our goal is to find p such that this whole expression equals 4. So, we set up the equation:

p² + 3p + 4 - (-5p + 84) = 4

This equation is the heart of our problem. Solving it will give us the value(s) of p that satisfy the condition. Remember, we are looking for a value of p that makes the difference between the two functions equal to 4.

Solving the Quadratic Equation

Now, let's simplify and solve this equation. First, distribute the negative sign:

p² + 3p + 4 + 5p - 84 = 4

Next, combine like terms:

p² + 8p - 80 = 4

To solve a quadratic equation, we usually want it in the form ax² + bx + c = 0. So, let's subtract 4 from both sides:

p² + 8p - 84 = 0

Okay, now we have a classic quadratic equation! There are a few ways to solve this: factoring, using the quadratic formula, or completing the square. Let's try factoring first. We need to find two numbers that multiply to -84 and add up to 8. After a bit of thought, we can see that 14 and -6 fit the bill:

(p + 14)(p - 6) = 0

This means that either (p + 14) = 0 or (p - 6) = 0. Solving these gives us two possible values for p:

p = -14 or p = 6

Checking the Interval and Solution

We've found two potential solutions, but remember, we're only interested in values of p within the interval [0, ∞). This means p must be greater than or equal to 0.

So, p = -14 is not a valid solution because it's negative. However, p = 6 is within our interval! This means that p = 6 is the value we're looking for.

Let's quickly verify our answer. We need to make sure that (f-g)(6) = 4. Plug in p = 6 into our functions:

f(6) = 6² + 3(6) + 4 = 36 + 18 + 4 = 58 g(6) = -5(6) + 84 = -30 + 84 = 54 (f-g)(6) = 58 - 54 = 4

Great! It checks out. So, the value of p that satisfies the condition is indeed 6.

Conclusion

So there you have it, folks! We successfully navigated through this problem by understanding the functions, setting up the equation, solving the quadratic, and checking our solution against the given interval. The value of p that makes (f-g)(p) = 4 on the interval [0, ∞) is p = 6. High five!

Let's move on to another math adventure, shall we?

Alright, guys, let's really dig deep into the concepts we used in the previous problem. We dealt with functions, intervals, and solving equations. Understanding these topics thoroughly will make you a math whiz in no time! We'll break it down step-by-step, using friendly language and real-world examples.

What Exactly is a Function?

Think of a function as a machine. You feed it an input (usually a number, represented by x), and it spits out an output (usually another number, represented by f(x)). The function is the rule that determines how the input is transformed into the output.

Key keywords to remember: input, output, rule, f(x), domain, range.

For example, in the function f(x) = x², the rule is "square the input." If you input x = 2, the function squares it and outputs f(2) = 4. If you input x = -3, the output is f(-3) = 9. See how it works?

The set of all possible inputs is called the domain of the function. The set of all possible outputs is called the range.

Functions can be represented in different ways:

• Equations: Like f(x) = x² + 3x + 4 (the one we used earlier!).
• Graphs: Visual representations of the relationship between inputs and outputs.
• Tables: Listing inputs and their corresponding outputs.
• Words: Describing the rule in plain English (e.g., "Multiply the input by 2 and add 1").

Understanding Intervals: Your Math Playground

An interval is simply a set of numbers between two endpoints. It's like a specific area on the number line where we're allowed to play. Intervals are often written using brackets and parentheses.

Key keywords to remember: endpoints, brackets, parentheses, bounded, unbounded, closed interval, open interval.

• Closed Interval: Includes the endpoints. Represented with square brackets [ ]. For example, [0, 5] means all numbers between 0 and 5, including 0 and 5.
• Open Interval: Does not include the endpoints. Represented with parentheses ( ). For example, (0, 5) means all numbers between 0 and 5, excluding 0 and 5.
• Half-Open (or Half-Closed) Interval: Includes one endpoint but not the other. For example, [0, 5) includes 0 but not 5, and (0, 5] includes 5 but not 0.
• Unbounded Intervals: Extend to infinity. We use the infinity symbol (∞) to represent them. For example, [0, ∞) means all numbers greater than or equal to 0.

In our original problem, we worked with the interval [0, ∞), which meant we were only looking for solutions that were non-negative numbers. This is a classic example of how intervals limit the possible values we need to consider.

Solving Equations: The Detective Work of Math

Solving an equation is like being a detective. We have a mystery – an unknown value (usually represented by a variable like x or p) – and we need to find it. Equations are mathematical statements that say two expressions are equal.

Key keywords to remember: variable, expression, equality, isolate, solution.

Our goal when solving an equation is to isolate the variable on one side of the equation. This means getting the variable all by itself, with no other terms or numbers attached to it. We do this by performing the same operations on both sides of the equation.

For example, let's say we have the equation:

2x + 3 = 7

To isolate x, we first subtract 3 from both sides:

2x = 4

Then, we divide both sides by 2:

x = 2

So, the solution to the equation is x = 2. We found the value of the variable that makes the equality true.

Quadratic Equations: A Special Case

Quadratic equations are equations of the form ax² + bx + c = 0, where a, b, and c are constants (numbers). These equations are a bit more challenging to solve than linear equations (like the one above).

Key keywords to remember: quadratic, factoring, quadratic formula, roots, zero product property.

There are a few main methods for solving quadratic equations:

• Factoring: If we can factor the quadratic expression into two linear expressions, we can use the zero product property, which says that if the product of two factors is zero, then at least one of the factors must be zero. This is what we did in our original problem.

• Quadratic Formula: A general formula that always works, even when factoring is difficult or impossible. The formula is:

  x = (-b ± √(b² - 4ac)) / (2a)

• Completing the Square: Another method that can be used to solve any quadratic equation.

Quadratic equations can have two solutions, one solution, or no real solutions (solutions that are not real numbers).

Putting It All Together: Real-World Applications

Okay, now you've got a solid understanding of functions, intervals, and solving equations. But how is this stuff actually used in the real world? You might be surprised!

• Physics: Functions are used to describe the motion of objects, the behavior of light and sound, and many other physical phenomena. Intervals are used to specify the range of values for physical quantities like time or distance.
• Engineering: Engineers use functions to design bridges, buildings, and other structures. They use equations to calculate stresses, strains, and other important parameters.
• Economics: Economists use functions to model supply and demand, predict market trends, and analyze economic data. Intervals can represent price ranges or time periods.
• Computer Science: Functions are the building blocks of computer programs. Intervals are used to specify ranges of data or time intervals for processing.

So, the concepts we've discussed are not just abstract mathematical ideas – they're powerful tools that can be used to solve real-world problems. By understanding functions, intervals, and solving equations, you're building a strong foundation for success in many fields.

Conclusion: Keep Exploring!

We've covered a lot of ground in this deep dive! We explored the world of functions, learned about intervals, and mastered the art of solving equations, especially quadratic equations. Remember, math is like a language – the more you practice, the more fluent you'll become. Keep exploring, keep asking questions, and keep challenging yourself. You've got this!

Alright, let's get practical and break down exactly how to tackle problems like the one we started with, where we need to find a value p that satisfies the equation (f-g)(p) = 4. This step-by-step guide will give you a clear roadmap to follow.

Key keywords for this section: step-by-step, procedure, f(p), g(p), simplify, quadratic equation, solutions, interval, verify.

Step 1: Understand the Problem

Before you even start writing anything down, make sure you fully grasp what the problem is asking. Read the problem carefully, identify the functions f(x) and g(x), and understand the interval in which you're looking for the solution. In our case, we had f(x) = x² + 3x + 4, g(x) = -5x + 84, and the interval [0, ∞).

Ask yourself:

• What are the given functions?
• What is the equation I need to solve?
• What is the interval for the solution?

Step 2: Set Up the Equation (f-g)(p) = 4

This is where we start translating the problem into mathematical language. Remember that (f-g)(p) means f(p) - g(p). So, replace x with p in the function definitions and write out the equation:

f(p) - g(p) = 4

Substitute the expressions for f(p) and g(p):

(p² + 3p + 4) - (-5p + 84) = 4

This step is crucial. Make sure you substitute correctly and pay attention to signs, especially when dealing with subtractions.

Step 3: Simplify the Equation

Now, it's time to clean up the equation and get it into a more manageable form. Distribute any negative signs and combine like terms:

p² + 3p + 4 + 5p - 84 = 4 p² + 8p - 80 = 4

Next, if necessary, rearrange the equation to get it into a standard form, like the quadratic form ax² + bx + c = 0. In our case, we subtract 4 from both sides:

p² + 8p - 84 = 0

The goal here is to simplify the equation as much as possible so that it's easier to solve.

Step 4: Solve for p

This is the heart of the problem! You need to find the value(s) of p that satisfy the equation. The method you use will depend on the type of equation you have. In our example, we have a quadratic equation, so we can use factoring, the quadratic formula, or completing the square.

We factored the equation as:

(p + 14)(p - 6) = 0

This gave us two potential solutions:

p = -14 or p = 6

Remember, choose the method that you're most comfortable with and that seems most appropriate for the equation.

Step 5: Check the Interval

This is a critical step that many people forget! You need to make sure that the solutions you found are actually within the given interval. In our case, the interval was [0, ∞), meaning p must be greater than or equal to 0.

We found p = -14 and p = 6. Since -14 is negative, it's not in the interval [0, ∞). However, 6 is greater than 0, so it's a valid solution.

Discard any solutions that are not within the specified interval. The interval acts as a filter, ensuring we only keep the solutions that make sense in the context of the problem.

Step 6: Verify the Solution (Optional but Recommended)

It's always a good idea to double-check your work, especially in math! To verify your solution, plug it back into the original equation (f-g)(p) = 4 and see if it holds true.

We plugged in p = 6 and found that (f-g)(6) = 4, so our solution is correct.

This step helps catch any errors you might have made along the way and gives you confidence in your answer.

Step 7: State the Answer Clearly

Finally, clearly state your answer in a way that's easy to understand. For example, you could say:

"The value of p that satisfies the equation (f-g)(p) = 4 on the interval [0, ∞) is p = 6."

Make sure you include all the relevant information in your answer, such as the variable you solved for, the equation, and the interval.

Conclusion: Practice Makes Perfect!

That's it! By following these steps, you can confidently solve problems of this type. Remember, math is a skill that improves with practice. The more problems you work through, the better you'll become at recognizing patterns, choosing the right methods, and avoiding common mistakes. So, grab some practice problems and start solving! You've got this!