Range Of Composite Function: G(f(x)) With E^x And X-2

Hey guys! Let's dive into a super interesting problem involving composite functions. We've got two functions here: $f(x) = e^x$ and $g(x) = x - 2$. Our mission, should we choose to accept it (and we do!), is to figure out the range of the composite function $(g ext{ o } f)(x)$. In simpler terms, we want to know all the possible output values of this combined function. Sounds like fun, right? Let's break it down step by step.

Understanding the Functions

First, let's get familiar with our players. We have $f(x) = e^x$, which is the exponential function with base e (Euler's number, approximately 2.718). The key characteristic of this function is that its output is always positive. No matter what real number you plug in for x, $e^x$ will always be greater than zero. Think about it: even if x is a large negative number, $e^x$ just gets closer and closer to zero, but never actually reaches it. If x is zero, $e^0$ is 1. And if x is a positive number, $e^x$ grows exponentially. This positivity is crucial for understanding the range of the composite function later on.

Next, we have $g(x) = x - 2$. This is a linear function, and it's a straightforward one. It simply takes any input x and subtracts 2 from it. So, if you give it 5, it spits out 3; if you give it 0, it gives you -2; and so on. Linear functions like this have a range of all real numbers, meaning they can output any number from negative infinity to positive infinity. However, the range of g(x) will be affected by the input it receives from f(x) in our composite function.

Forming the Composite Function (g o f)(x)

Now, let's get to the heart of the matter: the composite function $(g ext{ o } f)(x)$. Remember, this notation means that we first apply the function f to x, and then we take the result and plug it into the function g. So, we are essentially feeding the output of f(x) into g(x). Mathematically, we can write this as:

$(g ext{ o } f)(x) = g(f(x))$

To find the expression for $(g ext{ o } f)(x)$, we substitute f(x) into g(x) wherever we see x. Since $f(x) = e^x$ and $g(x) = x - 2$, we get:

$(g ext{ o } f)(x) = g(e^x) = e^x - 2$

Okay, now we have a new function: $(g ext{ o } f)(x) = e^x - 2$. This is the function whose range we need to determine. This is the critical step where we combine our understanding of both f(x) and g(x).

Determining the Range of (g o f)(x)

So, how do we find the range of $(g ext{ o } f)(x) = e^x - 2$? This is where our earlier observations about the range of $f(x) = e^x$ come into play. We know that $e^x$ is always greater than 0. It can get infinitely close to 0, but it never actually reaches it, and it can grow infinitely large as x increases.

Think of it this way: the smallest value $e^x$ can approach is 0 (but not include 0). If we then subtract 2 from this value, the smallest $(g ext{ o } f)(x)$ can approach is $0 - 2 = -2$. However, since $e^x$ never actually equals 0, $(g ext{ o } f)(x)$ will never actually equal -2. It will only get infinitely close to it.

On the other hand, as x gets larger and larger, $e^x$ also gets larger and larger, and so does $e^x - 2$. There's no upper bound to how large it can become. Therefore, $(g ext{ o } f)(x)$ can take on any value greater than -2.

We can express this mathematically as:

$(g ext{ o } f)(x) > -2$

This means the range of $(g ext{ o } f)(x)$ is all real numbers greater than -2. Understanding this inequality is the key to solving the problem.

Choosing the Correct Option

Now let's look back at the options we were given:

A. $y < 0$ B. $y > 0$ C. $y > -2$ D. all real numbers

Based on our analysis, the correct answer is C. $y > -2$. This is the only option that accurately describes the range of our composite function $(g ext{ o } f)(x)$.

Visualizing the Range (Optional)

If you're a visual learner, it can be helpful to think about the graph of $y = e^x - 2$. The graph of $y = e^x$ is a curve that starts very close to the x-axis (but never touches it) on the left and rises rapidly as x increases. Subtracting 2 from the function simply shifts the entire graph down by 2 units. So, the graph of $y = e^x - 2$ will look similar, but it will approach the horizontal line $y = -2$ instead of the x-axis. This visual representation reinforces the idea that the range is all values greater than -2.

Key Takeaways

Let's quickly recap the key steps we took to solve this problem:

1. Understood the individual functions: We analyzed the properties of $f(x) = e^x$ (always positive) and $g(x) = x - 2$ (linear function).
2. Formed the composite function: We found the expression for $(g ext{ o } f)(x)$ by substituting f(x) into g(x).
3. Determined the range: We used our knowledge of the range of $e^x$ to deduce the range of $e^x - 2$.
4. Chose the correct option: We selected the answer that matched our calculated range.

By breaking down the problem into smaller, manageable steps, we were able to tackle it effectively. Remember, understanding the properties of the individual functions is crucial for finding the range of a composite function.

Practice Makes Perfect

Composite functions can seem a bit tricky at first, but with practice, you'll become a pro in no time! Try working through similar problems with different functions. For example, you could try finding the range of $(f ext{ o } g)(x)$ instead of $(g ext{ o } f)(x)$, or you could use different functions for f(x) and g(x). The more you practice, the more comfortable you'll become with these types of problems. Don't be afraid to experiment and make mistakes – that's how we learn!

So, there you have it, guys! We've successfully navigated the world of composite functions and found the range of $(g ext{ o } f)(x)$. Keep up the great work, and I'll see you in the next problem!