Solving For P: A Step-by-Step Guide

by Alex Johnson 36 views

Hey there, math enthusiasts! Today, we're diving into a fun problem involving functions and a bit of algebraic manipulation. We're given two functions, f(x) and g(x), and a crucial piece of information: f(1/3) = g(1/3). Our mission? To find the value of p, a constant lurking within the g(x) function. Let's break down this problem step by step, making sure everything is clear and easy to follow. Remember, understanding the process is key! So, grab your pencils and let's get started!

Understanding the Problem: The Functions Unveiled

First things first, let's take a good look at the functions we're dealing with. We have:

  • f(x) = (9x/2)^-1
  • g(x) = √(1 - px^3), where p is a constant.

The notation might look a little intimidating at first, but don't worry! It's all about following the rules and taking things one step at a time. The key here is to understand what each part of the function means. f(x) essentially tells us that we take an input, x, perform a calculation, and get an output. In this case, we multiply x by 9/2 and then take the reciprocal (because of the -1 exponent). g(x), on the other hand, involves a square root. This function takes an input, x, cubes it, multiplies it by p, subtracts the result from 1, and then finds the square root of the whole thing. The trick is to follow the order of operations! The problem has provided a clue by letting us know that when x is 1/3, the output of the two functions is the same. That's our golden ticket to solving for p.

Breaking Down f(x)

Let's start by looking at f(x) = (9x/2)^-1. This function tells us to take an input x, multiply it by 9/2, and then raise the result to the power of -1. Remember, raising something to the power of -1 is the same as taking its reciprocal. So, we can rewrite f(x) as f(x) = 2/(9x). Much cleaner, right? This form makes it easier to substitute values and perform calculations. Now that we have a simplified f(x), we can plug in x = 1/3 and find the corresponding output value. This is a crucial step! The outcome that we obtain will be critical to the calculation of p when combined with g(x).

Demystifying g(x)

Now, let's turn our attention to g(x) = √(1 - px^3). This function is a bit more complex, but we'll tackle it step by step. Notice that p is a constant we are trying to find. This means that p is just a number! g(x) involves a cube operation and a square root, which might seem tricky at first, but with a bit of practice, you'll become a pro at these functions. Here, we see x being cubed and multiplied by p. Then, this result is subtracted from 1. Finally, the square root of the whole expression is taken. Remember that the value of x will be substituted with the value of 1/3, so that we can arrive at the value of p by using the value of f(1/3).

The Substitution Game: Finding f(1/3) and g(1/3)

Now, let's use the given information f(1/3) = g(1/3) to solve for p. This equation says that if we plug in x = 1/3 into both f(x) and g(x), we'll get the same output value for both. This will give us an equation that we can solve. It is important to know the order of operation and how to handle fractional exponents to solve this part of the problem. That's when we substitute x = 1/3 into both the f(x) and g(x) functions. This is the heart of the problem!

Calculating f(1/3)

Let's go back to our simplified f(x) = 2/(9x). Now, we'll substitute x = 1/3: f(1/3) = 2 / (9 * (1/3)) This looks straightforward! Now, let's do the math: f(1/3) = 2 / 3 . So, f(1/3) = 2/3. Now, we have a numerical value for f(1/3). This means that we know the output of f(x) when x = 1/3. That's great! It's one piece of the puzzle to finding p. Remember this number – it's going to be essential for the next step. Take your time with these calculations and double-check your work.

Setting up g(1/3)

Now, let's find the expression for g(1/3). We start with g(x) = √(1 - px^3). Substitute x = 1/3: g(1/3) = √(1 - p * (1/3)^3). This is also looking good! We've successfully substituted x in the function g(x). We can simplify this a little bit further to get g(1/3) = √(1 - p/27). We know that f(1/3) = g(1/3), so we can now set up an equation.

The Big Equation: Solving for p

Here comes the exciting part! Since we know that f(1/3) = g(1/3), and we've calculated values for both, we can set up an equation: 2/3 = √(1 - p/27). This equation is the key to finding p! It brings everything together. We are one step away from solving for p. We have an equation where the only unknown variable is p. This is great! The rest is just algebra, and it becomes easier to solve. Now, let's solve for p!

Isolating p: The Final Push

To solve for p, we need to isolate it. First, we need to get rid of that pesky square root. We can do that by squaring both sides of the equation. So, (2/3)^2 = (√(1 - p/27))^2. This simplifies to 4/9 = 1 - p/27. Then, we can subtract 1 from both sides to get 4/9 - 1 = -p/27. This becomes -5/9 = -p/27. Next, to isolate p, we can multiply both sides by -27. So, (-5/9) * -27 = p. This gives us p = 15. We have our answer! The constant p is equal to 15. The final answer should be checked by placing the value of p into the function and working backward to see if the equation holds true!

The Grand Finale: Conclusion and Takeaways

And there you have it! Through careful substitution and some algebraic manipulation, we successfully found the value of p. We started with two functions, used the given information f(1/3) = g(1/3), and worked our way through each step, making sure to show every calculation. The key takeaways from this problem are:

  • Understanding function notation and what each part of a function represents.
  • Being comfortable with substitution and following the order of operations.
  • The ability to manipulate equations and isolate variables.

This type of problem is a great way to practice your algebra skills and reinforce your understanding of functions. Math is all about practice and understanding. The more you work through problems like this, the more confident you'll become. So, keep practicing, keep learning, and keep enjoying the world of mathematics!

Conclusion

In conclusion, we successfully determined the value of p to be 15 by carefully evaluating the functions f(x) and g(x) and applying the given condition f(1/3) = g(1/3). This exercise underscores the importance of precision in algebraic manipulation and a solid grasp of function composition. Remember to always double-check your calculations and to break down complex problems into manageable steps. This approach not only helps you find the correct answer but also deepens your understanding of the underlying mathematical concepts. Keep practicing, and you'll find that solving these types of problems becomes easier and more enjoyable over time!

For further information on functions and algebra, you can check out this trusted resource: Khan Academy. They have amazing resources and examples. I recommend it to everyone! Good luck, and keep practicing!