Composite Function: Find (f ∘ G)(1) Simply!

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Composite Function: Find (f ∘ g)(1) Simply!

Hey guys! Today, let's dive into a fun little problem involving composite functions. We're given two functions, f(x)=5x−1f(x) = 5x - 1 and g(x)=x2−4g(x) = x^2 - 4, and our mission is to find the value of $(f "), what is the value of the composite function $(f "), what is the value of (f∘g)(1)(f \circ g)(1). Don't worry, it's easier than it sounds! We'll break it down step by step so you can totally nail it. So, grab your pencils, and let's get started!

Understanding Composite Functions

Before we jump into the problem, let's quickly recap what composite functions are all about. A composite function is basically a function within a function. When we write $(f "), what is the value of the composite function $(f "), what is the value of (f∘g)(x)(f \circ g)(x), it means we're plugging the function g(x)g(x) into the function f(x)f(x). In other words, we first evaluate g(x)g(x), and then we take the result and plug it into f(x)f(x). It's like a chain reaction of functions!

Think of it like this: imagine g(x)g(x) is a machine that takes an input xx and spits out a new value. Then, f(x)f(x) is another machine that takes that new value as its input and produces a final result. The composite function $(f "), what is the value of the composite function $(f "), what is the value of (f∘g)(x)(f \circ g)(x) represents the entire process of running the input xx through both machines, first g(x)g(x) and then f(x)f(x).

To evaluate a composite function at a specific value, like $(f "), what is the value of the composite function $(f "), what is the value of (f∘g)(1)(f \circ g)(1), we first evaluate the inner function g(1)g(1), and then we plug that result into the outer function f(x)f(x). This step-by-step approach makes evaluating composite functions straightforward and manageable. Remember, the order matters! $(f "), what is the value of the composite function $(f "), what is the value of (f∘g)(x)(f \circ g)(x) is generally not the same as $(g "), what is the value of the composite function $(g "), what is the value of (g∘f)(x)(g \circ f)(x), so pay close attention to the order in which the functions are composed.

Step-by-Step Solution

Now that we've refreshed our understanding of composite functions, let's tackle the problem at hand. We need to find $(f "), what is the value of the composite function $(f "), what is the value of (f∘g)(1)(f \circ g)(1), given f(x)=5x−1f(x) = 5x - 1 and g(x)=x2−4g(x) = x^2 - 4. Here's how we'll do it:

Step 1: Evaluate the Inner Function, g(1)

First, we need to find the value of g(1)g(1). We do this by substituting x=1x = 1 into the expression for g(x)g(x):

g(1)=(1)2−4=1−4=−3g(1) = (1)^2 - 4 = 1 - 4 = -3

So, g(1)=−3g(1) = -3. This means that when we plug 1 into the function g(x)g(x), we get -3 as the output. This value will now become the input for our next step, where we'll use the function f(x)f(x).

Step 2: Evaluate the Outer Function, f(g(1))

Now that we know g(1)=−3g(1) = -3, we can plug this value into the function f(x)f(x). In other words, we need to find f(−3)f(-3). We substitute x=−3x = -3 into the expression for f(x)f(x):

f(−3)=5(−3)−1=−15−1=−16f(-3) = 5(-3) - 1 = -15 - 1 = -16

Therefore, f(−3)=−16f(-3) = -16. This tells us that when we input -3 into the function f(x)f(x), the output is -16. Since -3 was the result of g(1)g(1), this means that $(f "), what is the value of the composite function $(f "), what is the value of (f∘g)(1)=−16(f \circ g)(1) = -16.

Step 3: State the Final Answer

We've done it! We found that $(f "), what is the value of the composite function $(f "), what is the value of (f∘g)(1)=−16(f \circ g)(1) = -16. This is the value of the composite function when we plug 1 into g(x)g(x) and then plug the result into f(x)f(x).

Alternative Approach: Finding the Composite Function First

Instead of evaluating the functions step-by-step, we could first find the general expression for the composite function $(f "), what is the value of the composite function $(f "), what is the value of (f∘g)(x)(f \circ g)(x) and then substitute x=1x = 1. Let's see how that works:

Step 1: Find the Expression for (f ∘ g)(x)

To find $(f "), what is the value of the composite function $(f "), what is the value of (f∘g)(x)(f \circ g)(x), we substitute g(x)g(x) into f(x)f(x) wherever we see xx:

$(f "), what is the value of the composite function $(f "), what is the value of (f∘g)(x)=f(g(x))=f(x2−4)=5(x2−4)−1(f \circ g)(x) = f(g(x)) = f(x^2 - 4) = 5(x^2 - 4) - 1

Now, we simplify the expression:

$(f "), what is the value of the composite function $(f "), what is the value of (f∘g)(x)=5x2−20−1=5x2−21(f \circ g)(x) = 5x^2 - 20 - 1 = 5x^2 - 21

So, the composite function $(f "), what is the value of the composite function $(f "), what is the value of (f∘g)(x)(f \circ g)(x) is 5x2−215x^2 - 21.

Step 2: Evaluate (f ∘ g)(1)

Now that we have the expression for the composite function, we can easily evaluate it at x=1x = 1:

$(f "), what is the value of the composite function $(f "), what is the value of (f∘g)(1)=5(1)2−21=5−21=−16(f \circ g)(1) = 5(1)^2 - 21 = 5 - 21 = -16

As you can see, we get the same answer as before: $(f "), what is the value of the composite function $(f "), what is the value of (f∘g)(1)=−16(f \circ g)(1) = -16.

Which Method is Better?

Both methods are perfectly valid for solving this problem. The step-by-step method is often easier to understand initially because it breaks down the problem into smaller, more manageable steps. You first evaluate the inner function and then use that result to evaluate the outer function. This approach can be particularly helpful when dealing with more complex composite functions.

The alternative method, where you first find the general expression for the composite function, can be more efficient if you need to evaluate the composite function at multiple values of xx. Once you have the expression for $(f "), what is the value of the composite function $(f "), what is the value of (f∘g)(x)(f \circ g)(x), you can simply plug in different values of xx to find the corresponding values of the composite function. However, this method can be more challenging if the functions are complex, as finding the general expression for the composite function might involve more complicated algebraic manipulations.

In general, the best method depends on the specific problem and your personal preference. If you're only evaluating the composite function at one value, the step-by-step method is often simpler. If you need to evaluate it at multiple values, finding the general expression for the composite function might be more efficient in the long run.

Practice Makes Perfect

Composite functions might seem a bit abstract at first, but with practice, they become much easier to handle. Try working through more examples and experimenting with different types of functions. The more you practice, the more comfortable you'll become with the concept of composite functions and the different techniques for evaluating them. So, keep practicing, and you'll be a composite function pro in no time!

Conclusion

So there you have it! We've successfully found that $(f "), what is the value of the composite function $(f "), what is the value of (f∘g)(1)=−16(f \circ g)(1) = -16. Remember, composite functions are all about plugging one function into another. Whether you evaluate step-by-step or find the general expression first, the key is to understand the order of operations and take your time. Keep practicing, and you'll master composite functions in no time. Happy calculating!