Composite Function: Find (f∘h∘g)(x) Explained!

Hey guys! Today, we're diving into the world of composite functions. Specifically, we're going to figure out how to find $(f \circ h \circ g)(x)$ when given the functions $f(x)=\frac{x+4}{x}$, $g(x)=x-2$, and $h(x)=4x-1$. Don't worry, it sounds more complicated than it actually is. Let's break it down step by step!

Understanding Composite Functions

Before we jump into the problem, let's quickly recap what a composite function is. A composite function is essentially a function within a function. When we write $(f \circ g)(x)$, it means we're plugging the function $g(x)$ into the function $f(x)$. In other words, $(f \circ g)(x) = f(g(x))$. The order matters! So, $(f \circ g)(x)$ is generally not the same as $(g \circ f)(x)$.

In our case, we have three functions composed together: $(f \circ h \circ g)(x)$. This means we're plugging $g(x)$ into $h(x)$, and then plugging the result into $f(x)$. So, $(f \circ h \circ g)(x) = f(h(g(x)))$. Remember this order, and you'll be golden!

Step-by-Step Solution

Now that we understand what we're dealing with, let's solve the problem. We're given:

• $f(x) = \frac{x+4}{x}$
• $g(x) = x-2$
• $h(x) = 4x-1$

Our goal is to find $(f \circ h \circ g)(x) = f(h(g(x)))$.

Step 1: Find h(g(x))

First, we need to find $h(g(x))$. This means we're plugging $g(x) = x-2$ into $h(x)$. So we replace every $x$ in $h(x)$ with $(x-2)$:

$h(g(x)) = h(x-2) = 4(x-2) - 1$

Now, let's simplify this expression:

$h(g(x)) = 4x - 8 - 1 = 4x - 9$

Great! So, $h(g(x)) = 4x - 9$.

Step 2: Find f(h(g(x)))

Next, we need to find $f(h(g(x)))$. We know that $h(g(x)) = 4x - 9$, so we need to plug this into $f(x)$. This means we replace every $x$ in $f(x)$ with $(4x - 9)$:

$f(h(g(x))) = f(4x - 9) = \frac{(4x - 9) + 4}{4x - 9}$

Now, let's simplify this expression:

$f(h(g(x))) = \frac{4x - 9 + 4}{4x - 9} = \frac{4x - 5}{4x - 9}$

And that's it! We've found $(f \circ h \circ g)(x)$.

Final Answer

Therefore, $(f \circ h \circ g)(x) = \frac{4x - 5}{4x - 9}$.

Common Mistakes to Avoid

• Order of Operations: The most common mistake is getting the order of the functions wrong. Remember, you're plugging the innermost function into the next, and so on. Always work from the inside out. Double-check your order to make sure you're not accidentally calculating something like $g(f(h(x)))$ instead. It's easy to get turned around, especially when you're dealing with three functions! Writing it out clearly can save you a lot of headaches. Use parentheses to clearly show which function is being plugged into which. This helps visually organize the process and reduces the chance of error. And when in doubt, rewrite the entire expression step-by-step. For example, instead of trying to do everything at once, write out $h(g(x))$ first, simplify it, and then plug that result into $f(x)$.
• Algebra Errors: Be careful when simplifying expressions. It's easy to make a mistake when distributing, combining like terms, or simplifying fractions. Pay close attention to signs (plus and minus) and make sure you're distributing correctly. A small error in the algebra can throw off the entire answer. When substituting, use parentheses to avoid sign errors, especially when dealing with negative numbers or expressions. Always double-check your work, especially after simplifying a complex expression.
• Forgetting to Simplify: Make sure you simplify your final answer as much as possible. Fully simplified answers are generally expected, so don't leave any terms that can be combined or fractions that can be reduced. After you've found the composite function, take a moment to see if there's anything you can simplify. This might involve combining like terms in the numerator or denominator, factoring, or canceling common factors. If you're unsure, try a few different simplification techniques to see if you can arrive at a simpler form.

Practice Problems

To really nail down this concept, try these practice problems:

1. If $f(x) = 2x + 1$, $g(x) = x^2$, and $h(x) = x - 3$, find $(f \circ g \circ h)(x)$.
2. If $f(x) = \frac{1}{x}$, $g(x) = 3x - 2$, and $h(x) = x + 1$, find $(f \circ h \circ g)(x)$.
3. If $f(x) = \sqrt{x}$, $g(x) = x + 5$, and $h(x) = 2x$, find $(f \circ g \circ h)(x)$.

Real-World Applications

While composite functions might seem abstract, they actually have real-world applications. For instance, consider a store that's offering a discount. The original price is a function, and the discount applied is another function. The final price you pay is the composite of these two functions. Similarly, in computer graphics, transformations like scaling, rotation, and translation are often represented as composite functions.

Conclusion

So, there you have it! Finding composite functions might seem tricky at first, but with practice, you'll become a pro. Just remember to work from the inside out, be careful with your algebra, and simplify your final answer. Good luck, and happy calculating! Remember to always double-check your work and practice consistently to master this concept. You got this!