Unlock Exponential Functions: Domain & Range Secrets

Jan 22, 2026 by Andrew McMorgan 53 views

Hey math whizzes and number crunchers! Today, we're diving deep into the fascinating world of exponential functions. You know, those awesome equations that grow or shrink super fast? We're going to break down three specific functions: $f(x)= rac{3}{2}(4)^x$ , $g(x)= rac{3}{2}(4)^{-x}$ , and $h(x)=- rac{3}{2}(4)^x$ . We'll be tackling the domain and range of each, and figuring out which statements about them are actually true. Get ready to level up your math game, guys!

Understanding Domain and Range in Exponential Functions

Before we get our hands dirty with these functions, let's quickly recap what domain and range mean in the context of functions. The domain is basically all the possible input values (the 'x' values) that a function can accept. Think of it as the set of ingredients you can put into your function recipe. The range, on the other hand, is all the possible output values (the 'y' or 'f(x)' values) that the function can produce. It's the delicious meals you can create with those ingredients.

Now, when we talk about exponential functions, they have some pretty predictable characteristics regarding their domain and range. For a basic exponential function of the form $y = a imes b^x$ (where 'b' is positive and not equal to 1), the domain is always all real numbers. This means you can plug in any number you can think of for 'x', and the function will happily spit out a result. This is because raising a positive base to any real power always yields a real number. Pretty cool, right? The range, however, depends on the value of 'a' and whether the base 'b' is greater than 1 or between 0 and 1. If 'a' is positive, the range is typically all positive real numbers ( $y > 0$ ). If 'a' is negative, the range is all negative real numbers ( $y < 0$ ). Why? Because $b^x$ is always positive, so multiplying it by a positive 'a' keeps it positive, and multiplying by a negative 'a' keeps it negative. The function never crosses the x-axis (the line $y=0$ ). It just gets infinitely close to it.

Let's take a look at our specific functions and see how these principles apply. We've got $f(x)= rac{3}{2}(4)^x$ , $g(x)= rac{3}{2}(4)^{-x}$ , and $h(x)=- rac{3}{2}(4)^x$ . We're going to dissect each one, paying close attention to their domains and ranges, and then we'll tackle those True/False statements. This is where the real fun begins, folks! We're not just memorizing rules; we're understanding why they work. It's all about building that solid mathematical foundation, and trust me, it makes tackling more complex problems down the line a total breeze. So, buckle up, grab your favorite thinking cap, and let's get this math party started!

Deconstructing $f(x)= rac{3}{2}(4)^x$

Alright, let's start with our first function, $f(x)= rac{3}{2}(4)^x$ . This looks like a pretty standard exponential function, right? We have a base of 4, which is greater than 1, so we expect some serious growth happening here. The coefficient $rac{3}{2}$ is positive. So, what can we say about its domain and range? As we discussed, for any function of the form $y = a imes b^x$ where $b > 0$ and $b eq 1$ , the domain is always all real numbers. There's no restriction on what 'x' can be. You can plug in 0, 1, -5, 100, $rac{ ext{pi}}{2}$ , literally any real number, and the function will give you a valid output. This is a fundamental characteristic of exponential functions. The base (4 in this case) raised to any real power results in a positive number. So, whether 'x' is positive, negative, or zero, $(4)^x$ will always be a positive value. When we multiply this positive result by our positive coefficient $rac{3}{2}$ , the output $f(x)$ will also always be positive. This brings us to the range. Since $(4)^x$ can take any positive value (approaching zero as x approaches negative infinity, and growing infinitely large as x approaches positive infinity), and we're multiplying it by a positive constant $rac{3}{2}$ , the output $f(x)$ will also take on all positive values. It will never be zero or negative. So, the range of $f(x)$ is all positive real numbers. We can write this as $f(x) > 0$ , or in interval notation as $(0, o)$ . This means the graph of $f(x)$ will always be above the x-axis, getting closer and closer to it as 'x' becomes a very large negative number, but never actually touching it. The y-intercept can be found by setting $x=0$ : $f(0) = rac{3}{2}(4)^0 = rac{3}{2}(1) = rac{3}{2}$ . This point $(0, rac{3}{2})$ is part of the graph. The fact that the domain is all real numbers is super important because it tells us we can explore the function's behavior across its entire horizontal extent. Similarly, understanding the range tells us about its vertical extent, giving us a clear picture of where the function's values lie. This foundational understanding is crucial as we move on to analyzing the other functions and evaluating the given statements. It’s like building blocks – each piece of knowledge supports the next, leading to a comprehensive grasp of these mathematical concepts. Remember, guys, the domain is about the 'x' possibilities, and the range is about the 'y' possibilities. Keep that distinction clear!

Examining $g(x)= rac{3}{2}(4)^{-x}$

Now, let's switch gears and look at $g(x)= rac{3}{2}(4)^{-x}$ . This function looks a bit different because of that negative sign in the exponent, $-x$ . But don't let it fool you! Remember that $(4)^{-x}$ is the same as $rac{1}{(4)^x}$ or $( rac{1}{4})^x$ . So, we can rewrite $g(x)$ as $g(x)= rac{3}{2}( rac{1}{4})^x$ . This is still an exponential function, but now our base is $rac{1}{4}$ , which is between 0 and 1. This means instead of growing rapidly, this function will be decaying rapidly as 'x' increases. Now, let's talk domain and range. Just like with $f(x)$ , the domain of $g(x)$ is all real numbers. There are no restrictions on the 'x' values we can input. We can plug in any real number, and the function will produce a valid output. The presence of the negative exponent doesn't change the fact that the base (whether we consider it 4 or $rac{1}{4}$ ) is positive, and any real number raised to a positive base (or any real number exponent applied to a positive base) results in a positive number. So, $4^{-x}$ will always be positive, and multiplying it by the positive coefficient $rac{3}{2}$ means that $g(x)$ will always be positive. Therefore, the range of $g(x)$ is all positive real numbers, just like $f(x)$ . We write this as $g(x) > 0$ , or in interval notation $(0, o)$ . The graph of $g(x)$ will also always be above the x-axis. The difference lies in its behavior: as 'x' gets larger and larger (approaches positive infinity), $g(x)$ gets closer and closer to 0. Conversely, as 'x' becomes a very large negative number (approaches negative infinity), $g(x)$ grows infinitely large. The y-intercept is found by setting $x=0$ : $g(0) = rac{3}{2}(4)^{-0} = rac{3}{2}(4)^0 = rac{3}{2}(1) = rac{3}{2}$ . So, the point $(0, rac{3}{2})$ is also on the graph of $g(x)$ . It's interesting to note that $g(x)$ is essentially a reflection of $f(x)$ across the y-axis. If you graph $f(x)$ and then reflect it over the y-axis, you get $g(x)$ . This reflection is caused by the change in the exponent from $x$ to $-x$ . Both functions share the same domain and range, but their growth/decay patterns are inverse. Understanding this symmetry and transformation is key to mastering function analysis. It shows how small changes in the function's equation can lead to significant, yet predictable, alterations in its graphical representation and behavior.

Analyzing $h(x)=- rac{3}{2}(4)^x$

Finally, let's tackle our third function, $h(x)=- rac{3}{2}(4)^x$ . This one has a twist – a negative sign right out front! This negative sign is going to flip our function upside down. Let's break down its domain and range. For the domain, as with the other exponential functions we've examined, there are no restrictions on the 'x' values. You can plug in any real number, and the function will compute a result. So, the domain of $h(x)$ is all real numbers. Now, for the range, things get interesting because of that leading negative sign. We know that $(4)^x$ will always produce a positive number. When we multiply this positive number by our negative coefficient $- rac{3}{2}$ , the result, $h(x)$ , will always be a negative number. It will never be zero or positive. This means the range of $h(x)$ is all negative real numbers. We can express this as $h(x) < 0$ , or in interval notation as $(- o, 0)$ . The graph of $h(x)$ will lie entirely below the x-axis. It will approach the x-axis as 'x' approaches positive infinity (getting closer to 0 from the negative side), and it will go down towards negative infinity as 'x' approaches negative infinity. The y-intercept is found by setting $x=0$ : $h(0) = - rac{3}{2}(4)^0 = - rac{3}{2}(1) = - rac{3}{2}$ . So, the point $(0, - rac{3}{2})$ is on the graph. Essentially, $h(x)$ is a reflection of $f(x)$ across the x-axis. If you take the graph of $f(x)$ and flip it vertically, you get the graph of $h(x)$ . This vertical reflection is directly caused by multiplying the entire function by -1. It's a powerful transformation that completely alters the vertical behavior of the function while leaving the horizontal possibilities (the domain) unchanged. This comparison between $f(x)$ and $h(x)$ really highlights how coefficients, especially their signs, play a crucial role in shaping the output and graphical representation of exponential functions. It's not just about the base; it's about every component of the function's definition.

Evaluating the Statements

Now that we've thoroughly analyzed $f(x)$ , $g(x)$ , and $h(x)$ , let's look back at the statements provided and see which ones are true. We need to consider the domain and range for each function.

Statement A: $f(x)$ has a domain of all real numbers.

Based on our analysis of $f(x)= rac{3}{2}(4)^x$ , we determined that the domain of any standard exponential function of this form is indeed all real numbers. We can plug in any real number for 'x' and get a valid output. So, Statement A is TRUE. This is a fundamental property of exponential functions.

Statement B: $g(x)$ has a range of all real numbers.

Let's recall our findings for $g(x)= rac{3}{2}(4)^{-x}$ . We found that the range of $g(x)$ is all positive real numbers ( $g(x) > 0$ ). This is because the term $(4)^{-x}$ (or $( rac{1}{4})^x$ ) is always positive, and multiplying it by the positive coefficient $rac{3}{2}$ keeps the output positive. The function never produces zero or negative values. Therefore, the statement that $g(x)$ has a range of all real numbers is FALSE. The range is restricted to positive values.

Statement C: $h(x)$ has a range of all negative real numbers.

Looking at $h(x)=- rac{3}{2}(4)^x$ , we saw that the leading negative sign flips the function downwards. Since $(4)^x$ is always positive, multiplying it by $- rac{3}{2}$ ensures that $h(x)$ is always a negative number. Thus, the range of $h(x)$ is all negative real numbers ( $h(x) < 0$ ). So, Statement C is TRUE.

Statement D: $f(x)$ has a range of all real numbers.

We previously established that the range of $f(x)= rac{3}{2}(4)^x$ is all positive real numbers ( $f(x) > 0$ ). It never goes below the x-axis. Therefore, the statement that $f(x)$ has a range of all real numbers is FALSE. Its range is restricted to positive values.

Statement E: $g(x)$ has a domain of all real numbers.

Similar to $f(x)$ , the function $g(x)= rac{3}{2}(4)^{-x}$ is an exponential function. The presence of the negative exponent does not restrict the possible input values for 'x'. We can substitute any real number for 'x' and get a valid output. Therefore, the domain of $g(x)$ is all real numbers. So, Statement E is TRUE.

Statement F: $h(x)$ has a domain of all real numbers.

And finally, for $h(x)=- rac{3}{2}(4)^x$ , the negative sign out front affects the range, not the domain. The base is still raised to the power of 'x', and 'x' can be any real number. Thus, the domain of $h(x)$ is also all real numbers. So, Statement F is TRUE.

Conclusion: Which Statements are True?

After careful examination of each function's domain and range, we can confidently conclude that the following statements are true:

Statement A: $f(x)$ has a domain of all real numbers.
Statement C: $h(x)$ has a range of all negative real numbers.
Statement E: $g(x)$ has a domain of all real numbers.
Statement F: $h(x)$ has a domain of all real numbers.

It's awesome how understanding the basic rules of exponential functions, combined with how transformations like negative exponents or leading negative signs affect them, allows us to accurately determine their domains and ranges. Keep practicing, and you'll become a domain and range master in no time! Keep exploring the world of math, guys – it's full of amazing discoveries!