Reflecting Exponential Functions: Finding G(x) And Its Values

Hey math enthusiasts! Today, we're diving into the world of exponential functions and transformations. Specifically, we'll explore what happens when we reflect a function across the x-axis. We'll start with a given function, perform the reflection, and then analyze the resulting function. Let's get started, shall we?

Understanding the Reflection of Exponential Functions

Okay, guys, let's break down the core concept here. When a function is reflected across the x-axis, every y-value (or the output) of the original function changes its sign. Think of it like a mirror image. If the original function has a point at (x, y), the reflected function will have a point at (x, -y). This transformation affects the entire function, flipping it upside down. This is important for understanding how to define g(x). Our starting function is $f(x) = 2(3.5)^x$. We need to figure out what $g(x)$ looks like after reflection. The key here is to recognize that we're changing the sign of the output. So, for every value of $f(x)$, the corresponding value in $g(x)$ will be the negative of that. In mathematical terms, this means that $g(x) = -f(x)$. Since $f(x) = 2(3.5)^x$, we can substitute that into our new equation, giving us $g(x) = -2(3.5)^x$. So, when we reflect $f(x) = 2(3.5)^x$ over the x-axis, we obtain the function $g(x) = -2(3.5)^x$. Now that we know how to write the function definition of g(x), let's find the initial value.

Defining the Reflected Function: The Function Definition of g(x)

Alright, let's talk about the function definition of $g(x)$. As we've established, the reflection across the x-axis negates the function's output. Therefore, if $f(x) = 2(3.5)^x$, the function $g(x)$, which is the reflection of $f(x)$, is defined as $g(x) = -2(3.5)^x$. It is super important to remember this rule: reflection across the x-axis means the function's value gets multiplied by -1. The core exponential component, $(3.5)^x$, remains the same, but the coefficient, which was originally 2, becomes -2. This changes the graph, it flips across the x-axis. So, to summarize, the correct way to write the function is to make sure we include the negative sign in front of the 2, so the final form becomes $g(x) = -2(3.5)^x$. This means that for every x-value, $g(x)$ will output the opposite sign of $f(x)$. The reflection is now fully defined. Pretty cool, right? This is the core concept of this transformation, and understanding it is key to the rest of the problem.

Determining the Initial Value of g(x)

Now, let's move on to the initial value of $g(x)$. The initial value of a function is the output when the input (x) is 0. To find this, we simply need to substitute $x = 0$ into our function $g(x) = -2(3.5)^x$. Let's do it: $g(0) = -2(3.5)^0$. Any non-zero number raised to the power of 0 equals 1. Therefore, $(3.5)^0 = 1$. This simplifies the equation to $g(0) = -2(1)$. Calculating this, we find that $g(0) = -2$. Thus, the initial value of $g(x)$ is -2. This means that the graph of $g(x)$ crosses the y-axis at the point (0, -2). This is where the function starts, and it's a key point to understand the behavior of the reflected function. Knowing the initial value is also useful for graphing and understanding the function's trend. The value -2 is very important. Since $f(0)$ was equal to 2, it is easy to see how the reflection changed the sign. Great job, guys! You have successfully found the initial value of $g(x)$.

Calculating Outputs for Specific Inputs: -1 and 1

Now, let's calculate the outputs of $g(x)$ for specific inputs: -1 and 1. This will give us a clearer understanding of how the function behaves at different points. First, let's find $g(-1)$. We plug in $x = -1$ into the function: $g(-1) = -2(3.5)^{-1}$. Remember that a negative exponent means we take the reciprocal of the base. Thus, 3.5^{-1} = rac{1}{3.5}. Now, let's calculate: g(-1) = -2 imes rac{1}{3.5}. This gives us g(-1) = -rac{2}{3.5}. Which simplifies to approximately -0.57. Next, let's find $g(1)$. We substitute $x = 1$ into the function: $g(1) = -2(3.5)^1$. Anything to the power of 1 is just itself, so $(3.5)^1 = 3.5$. This simplifies to $g(1) = -2 imes 3.5$, which means $g(1) = -7$. Thus, we see that $g(-1) ext{ is approximately } -0.57$ and $g(1) = -7$. These two points give us more information about the shape of the graph of $g(x)$, showing how it changes as x increases or decreases. This kind of calculation is very important to get a handle on the behavior of the reflected function.

Summarizing the Results and Understanding the Concepts

Alright, let's put everything together, guys! We started with the function $f(x) = 2(3.5)^x$ and reflected it across the x-axis, resulting in $g(x) = -2(3.5)^x$. The initial value of $g(x)$ is -2. For an input of -1, $g(-1) ext{ is approximately } -0.57$, and for an input of 1, $g(1) = -7$. Remember, the key takeaway is that reflecting a function across the x-axis changes the sign of its output. This changes the function. This is a fundamental concept in function transformations and is applicable to various types of functions, not just exponentials. Understanding this allows you to predict the behavior of transformed functions. The values we calculated show us that when x is negative, the value is close to zero, and as the positive values increase, the values decrease and become more negative, as the function values approach negative infinity. Understanding transformations of functions is a core concept in mathematics. By practicing problems like this, you can strengthen your understanding of exponential functions, their graphs, and how they behave under different transformations, like reflections. Keep practicing, and you'll become pros in no time. Keep the concepts of reflection and initial value in mind! You've got this!