Find G(x) Given Composite Function H(x) And F(x)

Hey guys, welcome back to Plastik Magazine! Today, we're diving deep into the fascinating world of mathematics, specifically tackling a super common yet sometimes tricky concept: composite functions. You know, the kind where one function is nested inside another, like $h(x) = f(g(x))$. We've got a specific problem here that's been bugging some of you, and we're going to break it down step-by-step. The question is: If $h(x) = \frac{4}{3} x - 1$ and we know that $h(x)$ is the result of $f(g(x))$, and we're also given that $f(x) = 2x - 1$, what exactly is $g(x)$? This might seem a bit like a puzzle, but trust me, once you get the hang of how these functions play together, it's actually pretty straightforward. We'll explore the different options provided (A, B, C, and D) and figure out which one correctly completes the equation. So, grab your thinking caps, maybe a calculator if you like, and let's unravel this mathematical mystery together!

Understanding Composite Functions: The "Inside-Out" Approach

Alright, let's get down to business. The core of this problem lies in understanding what $h(x) = f(g(x))$ actually means. Think of it like Russian nesting dolls, guys. You have an outer function, $f$, and an inner function, $g$. When you create the composite function $h(x)$, you take the entire expression for $g(x)$ and plug it into every 'x' that appears in the definition of $f(x)$. So, if $f(x)$ is defined as "take something, multiply it by 2, and then subtract 1", then $f(g(x))$ means "take the expression for $g(x)$, multiply that by 2, and then subtract 1". The result of this process is our function $h(x)$.

In our specific problem, we're given $h(x) = \frac{4}{3} x - 1$ and $f(x) = 2x - 1$. We need to find $g(x)$. This means we need to work backward, or perhaps more accurately, figure out what $g(x)$ must be so that when we plug it into $f(x)$, we get $h(x)$. The equation we're solving is essentially: $f( ext{our mystery } g(x)) = \frac{4}{3} x - 1$.

Let's represent our mystery function $g(x)$ as just '$g$' for a moment to keep things clean. So, we have $f(g) = \frac{4}{3} x - 1$. Since we know $f(x) = 2x - 1$, we can write $f(g)$ by substituting '$g$' for 'x' in the definition of $f(x)$: $f(g) = 2g - 1$. Now we have two expressions for $f(g)$: $2g - 1$ and $\frac{4}{3} x - 1$. By setting these equal to each other, we get our equation to solve for $g(x)$: $2g - 1 = \frac{4}{3} x - 1$. This is the key step! From here, it's just a matter of algebraic manipulation to isolate $g$ (which represents $g(x)$).

Don't forget that the goal is to find the expression for $g(x)$. We've set up the equation correctly by understanding the definition of a composite function. The next part is pure algebra, and it's crucial to be precise with your calculations. We'll walk through solving this equation in the next section, so stick with me!

Solving for g(x): The Algebraic Journey

Okay, so we've established the equation that $g(x)$ must satisfy: $2g(x) - 1 = \frac{4}{3} x - 1$. Our mission now is to isolate $g(x)$ on one side of the equation. This is where basic algebra skills come into play, and it's important to follow the order of operations correctly. First things first, let's get rid of that '-1' on both sides of the equation. If we add 1 to both sides, they cancel out:

$2g(x) - 1 + 1 = \frac{4}{3} x - 1 + 1$

This simplifies to:

$2g(x) = \frac{4}{3} x$

Now, $g(x)$ is being multiplied by 2. To get $g(x)$ by itself, we need to perform the inverse operation, which is division. We'll divide both sides of the equation by 2:

$\frac{2g(x)}{2} = \frac{\frac{4}{3} x}{2}$

On the left side, the 2s cancel out, leaving us with $g(x)$. On the right side, we have a fraction divided by a number. Remember that dividing by 2 is the same as multiplying by $\frac{1}{2}$. So, we can rewrite the right side as:

$g(x) = \frac{4}{3} x \times \frac{1}{2}$

Now, we multiply the numerators together and the denominators together:

$g(x) = \frac{4 \times 1}{3 \times 2} x$

$g(x) = \frac{4}{6} x$

Finally, we can simplify the fraction $\frac{4}{6}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

$g(x) = \frac{4 \div 2}{6 \div 2} x$

$g(x) = \frac{2}{3} x$

And there you have it! We've successfully isolated $g(x)$ and found its expression. This result, $g(x) = \frac{2}{3} x$, should be our answer. In the next section, we'll check this answer against the given options and briefly verify our work to make sure we haven't made any mistakes. This algebraic process is fundamental to solving many problems involving functions, so make sure you're comfortable with these steps!

Verifying the Solution and Choosing the Correct Option

So, we've arrived at our potential answer: $g(x) = \frac{2}{3} x$. Now, let's see if this actually works by plugging it back into the definition of $f(x)$ and checking if we get our original $h(x)$. Remember, $f(x) = 2x - 1$ and we want to calculate $f(g(x))$.

We substitute our found $g(x)$ into $f(x)$: $f(\frac{2}{3} x) = 2(\frac{2}{3} x) - 1$.

Now, perform the multiplication:

$f(g(x)) = 2 \times \frac{2}{3} x - 1$

$f(g(x)) = \frac{4}{3} x - 1$

And look at that! The result, $\frac{4}{3} x - 1$, is exactly the function $h(x)$ that was given in the problem. This confirms that our calculated $g(x)$ is indeed correct.

Now, let's look at the multiple-choice options provided:

A. $g(x)=\frac{2}{3} x$ B. $g(x)=\frac{2}{3} x-1$ C. $g(x)=2 \frac{1}{3} x$ D. $g(x)=2 \frac{1}{3} x-1$

Our derived function, $g(x) = \frac{2}{3} x$, perfectly matches option A. Options B, C, and D represent different functions that, if plugged into $f(x)$, would not yield the correct $h(x)$. For example, option B, $g(x)=\frac{2}{3} x-1$, would give $f(\frac{2}{3} x-1) = 2(\frac{2}{3} x-1) - 1 = \frac{4}{3} x - 2 - 1 = \frac{4}{3} x - 3$, which is not $h(x)$. Option C, $g(x)=2 \frac{1}{3} x = \frac{7}{3} x$, would result in $f(\frac{7}{3} x) = 2(\frac{7}{3} x) - 1 = \frac{14}{3} x - 1$, also incorrect. Option D combines the error of C with the subtraction term from B, leading to an even more incorrect result.

So, the correct answer is definitively A. $g(x)=\frac{2}{3} x$. It's always a good practice to double-check your work, especially when dealing with multiple-choice questions. We've successfully used the definition of composite functions and a bit of algebra to find the missing piece of the puzzle. Keep practicing these types of problems, guys, and you'll become function composition pros in no time!

Conclusion: Mastering Composite Functions

To wrap things up, solving for a component function within a composite function like $h(x) = f(g(x))$ involves a systematic approach. First, you need to deeply understand the definition of a composite function: $f(g(x))$ means you substitute the entire expression for $g(x)$ into the variable $x$ of the function $f(x)$. In our case, we were given the final composite function $h(x) = \frac{4}{3} x - 1$ and one of the original functions, $f(x) = 2x - 1$. Our goal was to find the other original function, $g(x)$.

We set up the core equation by knowing that $f(g(x))$ must equal $h(x)$. Since $f(x) = 2x - 1$, then $f(g(x))$ translates to $2(g(x)) - 1$. Equating this to the given $h(x)$, we got the algebraic equation $2g(x) - 1 = \frac{4}{3} x - 1$. The subsequent steps involved standard algebraic manipulation: adding 1 to both sides to simplify the equation to $2g(x) = \frac{4}{3} x$, and then dividing both sides by 2 to isolate $g(x)$. This led us to $g(x) = \frac{4}{3} x \div 2$, which simplifies to $g(x) = \frac{4}{3} x \times \frac{1}{2}$, ultimately resulting in $g(x) = \frac{4}{6} x$. Finally, we simplified the fraction $\frac{4}{6}$ to its lowest terms, $\frac{2}{3}$, giving us our answer: $g(x) = \frac{2}{3} x$.

We then took the crucial step of verifying our answer by substituting $g(x) = \frac{2}{3} x$ back into $f(x)=2x-1$. This yielded $f(\frac{2}{3} x) = 2(\frac{2}{3} x) - 1 = \frac{4}{3} x - 1$, which perfectly matched the given $h(x)$. This verification process is absolutely vital for ensuring accuracy, especially in mathematics. It confirms that option A, $g(x)=\frac{2}{3} x$, is the correct choice among the given options.

Mastering composite functions is a key skill in algebra and calculus. It requires a solid understanding of function notation, substitution, and algebraic problem-solving. By working through problems like this one, you build confidence and proficiency. Remember, the process is about deconstructing the problem, applying the correct definitions, solving the resulting equations, and verifying your findings. Keep pushing your mathematical boundaries, and don't hesitate to tackle more complex problems. Thanks for joining us on Plastik Magazine; we'll see you in the next article!