Domain & Range Of Exponential Function K(x) = -2^x

Hey guys, let's dive into the exciting world of exponential functions and figure out the domain and range of a specific function, $k(x) = -2^x$. We know that $k$ is a transformation of the parent exponential function, $f(x) = 2^x$. Understanding these transformations is key to unlocking the secrets of functions, so buckle up!

Understanding the Parent Function: f(x) = 2^x

Before we tackle $k(x) = -2^x$, it's super important to get a solid grip on its parent function, $f(x) = 2^x$. This is your basic, go-to exponential function. When we talk about the domain of $f(x) = 2^x$, we're asking ourselves: "What values of $x$ can we plug into this function without breaking it?" For exponential functions like this, there are no restrictions on the $x$-values. You can plug in any real number – positive, negative, zero, fractions, decimals, you name it! So, the domain of $f(x) = 2^x$ is all real numbers, which we can write as $(-\infty, \infty)$ or ${x \in R}$. Now, let's think about the range. The range is all about the possible output values, the $y$-values, that the function can produce. For $f(x) = 2^x$, since the base (2) is positive, the output will always be positive. No matter what real number you raise 2 to, the result will never be zero or negative. It will always be greater than zero. So, the range of $f(x) = 2^x$ is $(0, \infty)$ or ${y \in R \mid y > 0}$. Keep this in mind, as it's the foundation for understanding our transformed function.

Transforming the Parent Function: k(x) = -2^x

Alright, let's get down to business with our function $k(x) = -2^x$. This function is a transformation of the parent function $f(x) = 2^x$. Specifically, the negative sign in front of the $2^x$ indicates a reflection across the x-axis. Think about what this reflection does to the graph of $y = 2^x$. Every positive $y$-value from the parent function becomes a negative $y$-value in the transformed function. For instance, if $f(3) = 2^3 = 8$, then $k(3) = -2^3 = -8$. This transformation is crucial for determining the domain and range of $k(x)$.

Digging into the Domain of k(x) = -2^x

Now, let's nail down the domain of $k(x) = -2^x$. Remember, the domain is about the possible input values for $x$. Did our transformation (the reflection across the x-axis) add any restrictions to the values of $x$ we can plug in? Nope! Just like with the parent function $f(x) = 2^x$, we can still plug in any real number for $x$ into $k(x) = -2^x$. The negative sign only affects the output, not the input. So, the domain of $k(x)$ remains all real numbers. We can express this as $(-\infty, \infty)$ or ${x \in R}$. It's a common misconception that the negative sign might limit the domain, but in this case, it doesn't.

Unpacking the Range of k(x) = -2^x

The range is where the real magic of this transformation happens. The range refers to the set of all possible output $y$-values. We know that the parent function $f(x) = 2^x$ has a range of ${y \in R \mid y > 0}$. That means all its outputs are positive. When we apply the transformation $k(x) = -2^x$, which is a reflection across the x-axis, we are essentially taking all those positive $y$-values and making them negative. So, if $f(x)$ produced values like 1, 2, 4, 8, etc., then $k(x)$ will produce values like -1, -2, -4, -8, etc. This means that the output of $k(x)$ will always be negative. It will never be zero, and it will certainly never be positive. Therefore, the range of $k(x) = -2^x$ is all negative real numbers. We can write this as $(-\infty, 0)$ or ${y \in R \mid y < 0}$. This is a direct consequence of the reflection across the x-axis.

Comparing with the Options

Let's look at the options provided:

A. Domain: ${x \in R \mid -\infty < x < \infty}$, Range: ${y \in R \mid y \leq 0}$ B. Domain: ${x \in R \mid -\infty < x < \infty}$, Range: ${y \in R \mid y < 0}$

Based on our analysis, the domain of $k(x) = -2^x$ is indeed all real numbers, which is correctly represented as ${x \in R \mid -\infty < x < \infty}$.

Now for the range. We determined that the range consists of all negative real numbers, meaning $y$ must be strictly less than 0. Option A states the range as ${y \in R \mid y \leq 0}$, which includes 0. However, as we established, the function $k(x) = -2^x$ will never output 0. Option B states the range as ${y \in R \mid y < 0}$, which perfectly matches our findings. The outputs are all negative real numbers, excluding 0.

Conclusion: The Correct Answer

So, the correct domain and range for the function $k(x) = -2^x$ are:

• Domain: ${x \in R \mid -\infty < x < \infty}$ (all real numbers)
• Range: ${y \in R \mid y < 0}$ (all negative real numbers)

This means Option B is the correct answer, guys! Keep practicing these transformations, and you'll be a function master in no time. It's all about understanding how changes to the basic function affect its input and output possibilities. The reflection across the x-axis is a powerful transformation that flips the graph vertically, directly impacting the range by inverting all the output values. Remember this concept for future problems!