Domain Of Exponential Function Explained: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into a classic question about domains of functions, specifically looking at the function $f(x)=-\frac{5}{6}\left(\frac{3}{5}\right)^x$. This might seem a bit intimidating at first, but trust me, it's actually pretty straightforward. We'll break down what a domain is, and how to figure it out for this specific type of function. So, grab your pencils and let's get started! First of all, what does 'domain' even mean? In simple terms, the domain of a function is the set of all possible input values (usually x-values) for which the function is defined. Think of it like this: the domain is all the numbers you're allowed to 'plug in' to the function, and the function will actually give you a valid output. Conversely, any value not in the domain would cause the function to act up – maybe you'd get an undefined result, or some other kind of mathematical weirdness. The type of function we're dealing with here is an exponential function. Exponential functions have a base raised to a variable power. In our case, the base is $\frac{3}{5}$ and the exponent is x. The negative sign and the $\frac{5}{6}$ in front are just constants that affect the function's graph, but they don't impact the domain. So the core of what we're interested in is the $\left(\frac{3}{5}\right)^x$ part. To determine the domain, we need to consider if there are any restrictions on the values of x that we can use. Restrictions usually pop up when we have things like fractions with variables in the denominator (where we can't divide by zero), or square roots (where we can't take the square root of a negative number) and logarithms (where the argument of the logarithm must be positive). But in this case, we don't have any of those kinds of operations. The exponential part of the function, $\left(\frac{3}{5}\right)^x$, is defined for any real number x. This is because you can raise $\frac{3}{5}$ (or any positive number) to any power, and you'll always get a real number as a result. You can raise it to a positive power, a negative power, or even zero. So, there aren't any values of x that would cause the function to be undefined. The coefficient $-\frac{5}{6}$ doesn't change the domain because it's simply a constant multiplying the exponential term. Therefore, the domain of the function $f(x)=-\frac{5}{6}\left(\frac{3}{5}\right)^x$ is all real numbers. This means you can plug in any real number for x, and you'll get a valid output for f(x). This is a fundamental concept, so let's dive more into the mathematical specifics!

Diving Deeper: Understanding Exponential Functions and Their Domains

Alright, let's dig a little deeper into exponential functions to really solidify our understanding of their domains. Exponential functions are incredibly important in mathematics and show up in all sorts of real-world applications, from population growth to radioactive decay. Understanding their behavior is key, and the domain is a fundamental aspect of this understanding. So, let's reiterate the basic form of an exponential function: $f(x) = a imes b^x$, where: $a$ is a constant (a vertical scaling factor). $b$ is the base (a positive real number, not equal to 1, because $1^x$ would be just a constant function). x is the variable (the exponent). In our original function, $f(x)=-\frac{5}{6}\left(\frac{3}{5}\right)^x$, $a = -\frac{5}{6}$ and $b = \frac{3}{5}$. The x value is what we're concerned with for the domain. The crucial thing to remember about the base b is that it must be positive. Why? Because raising a negative number to fractional powers (like 1/2, for a square root) can lead to complex numbers. We want to stick with real numbers when determining the domain in this context, so the base has to be positive. However, the value of a can be any real number. This just scales the graph vertically and, as we saw, it doesn't affect the domain.

So, back to our original question. Because our base, $\frac{3}{5}$, is positive, we can raise it to any real power x. There are no restrictions on x that would make the function undefined. What about negative values of x? Remember that a negative exponent means we take the reciprocal of the base raised to the positive version of the exponent. For example, $3^{-2} = \frac{1}{3^2} = \frac{1}{9}$. This still results in a real number, and it's perfectly fine for our function. What about zero? Well, anything (except zero) raised to the power of zero is 1. So, $\left(\frac{3}{5}\right)^0 = 1$, which is perfectly valid. Thus, the domain includes all real numbers, positive, negative, and zero. This lack of restrictions is what makes exponential functions so versatile. You can plug in any real number, and you'll get a real number output. If you were to graph this function, you'd see a smooth curve that extends infinitely to the left and right along the x-axis, never hitting any 'gaps' where the function isn't defined. This characteristic of having a domain of all real numbers is typical for exponential functions with a positive base. This also helps in a variety of situations. For instance, in problems involving continuous compounding interest, we'd use an exponential function. Or in situations of radioactive decay, it is modeled through exponential functions as well. Having a clear understanding of the domain helps us ensure that our models are valid for all relevant input values. Now, let's go through the multiple choice to make sure our answer is correct.

Analyzing the Multiple-Choice Options

Okay, now that we've figured out the domain of our exponential function $f(x)=-\frac{5}{6}\left(\frac{3}{5}\right)^x$, let's go through the multiple-choice options to confirm our answer. We'll break down each choice and explain why it's correct or incorrect. A. all real numbers: This is our answer! As we discussed, there are no restrictions on the values of x that we can plug into the function. You can use any real number, and you'll get a valid output. The exponential part of the function, $\left(\frac{3}{5}\right)^x$, is defined for all real numbers, and the constants don't change the domain. B. all real numbers less than 0: This option is incorrect. This would mean you can only use negative numbers as input for x. However, we know that the function is defined for all real numbers, including positive numbers and zero. If you were to only have values less than 0, the graph will be on a specific range. C. all real numbers greater than 0: This is also incorrect. Similar to option B, this would limit our input to only positive numbers. The function is defined for negative numbers and zero as well. The curve would change. D. all real numbers less than or equal to 0: This option is incorrect for the same reasons as options B and C. It restricts the domain to negative numbers and zero, excluding all positive real numbers, which are perfectly valid inputs for our function. In summary, we've determined that the domain of the function $f(x)=-\frac{5}{6}\left(\frac{3}{5}\right)^x$ is all real numbers (option A). The exponential function is defined for any real number input. This understanding is important for solving more complex mathematical problems that involve this function. So there you have it! Keep practicing and you will master this type of questions. Remember that understanding the underlying concepts is always the key. Keep up the good work!