Understanding Domain And Range Of Logarithmic Functions

Hey math enthusiasts! Today, we're diving into the fascinating world of logarithmic functions, specifically tackling the concepts of domain and range. Understanding these two terms is super crucial for grasping the behavior of functions and where they "live" on the graph. So, let's break down the problem: "What are the domain and the range of this function?" $f(x)=\log (x+3)-2$. We'll explore what domain and range mean, how to find them for logarithmic functions, and then apply this knowledge to solve the given problem. Get ready to flex those math muscles!

Demystifying Domain and Range

Alright, first things first: let's get clear on what domain and range actually are. Think of a function like a machine. You put something in (the input), and it spits something out (the output). The domain is the set of all possible inputs you can feed into the machine – the values of x that the function can accept without breaking. The range, on the other hand, is the set of all possible outputs the machine can produce – the values of f(x) that the function can generate. It is all the possible y-values. Think of it like this: the domain is all the x-values, and the range is all the y-values. When it comes to functions, understanding domain and range helps us to understand where a function is defined and what values it can take. It's like knowing the ingredients (domain) and the final dish (range) of a recipe. Let's illustrate with an example.

Imagine a function f(x) = x + 2.

For the domain, since we can plug in any number for x and get a valid output, the domain is all real numbers (from negative infinity to positive infinity). This is usually written as (-∞, ∞).

For the range, the output f(x) will also cover all real numbers since for every value of x we plug in, we will get a value for f(x). So the range is also (-∞, ∞).

Now, let's consider a function with a restriction. Say f(x) = √x.

For the domain, we can only plug in non-negative numbers because we can't take the square root of a negative number (at least, not in the real number system). Therefore, the domain is all real numbers greater than or equal to zero, which is written as [0, ∞).

For the range, the output of the square root function will also only be non-negative values. So the range is also [0, ∞).

This simple example shows that some functions have restrictions on their inputs, and these restrictions define the domain. The domain dictates what kind of x values are allowed, while the range gives us the spread of the y values the function will actually produce. These definitions are fundamental to understanding functions, and they become even more interesting when we start dealing with special functions, such as logarithmic functions.

Unveiling the Domain of Logarithmic Functions

Okay, now let's focus on logarithmic functions. These functions are defined as f(x) = logₐ(x), where a is the base of the logarithm (and a is positive and not equal to 1). The key thing to remember about logarithms is that you can only take the logarithm of a positive number. You cannot take the logarithm of zero or a negative number. That's where the domain restrictions come in. The argument (the part inside the logarithm) must always be greater than zero. So, to find the domain of a logarithmic function, you need to set the argument greater than zero and solve for x.

Let’s apply this rule: Consider the function f(x) = log(x + 3) - 2.

We need to find the values of x that make the expression inside the logarithm, (x + 3), positive. So, we set up the inequality x + 3 > 0. Solving for x, we get x > -3. This tells us that the domain of the function is all real numbers greater than -3. In interval notation, this is written as (-3, ∞). That means you can plug in any number greater than -3 into this function, and you'll get a real number as an output. Any number less than or equal to -3 would break the function, since you can't take the log of a non-positive number.

Understanding the domain is really about understanding the limitations of the function. For logarithmic functions, these limitations are dictated by the fact that the argument must be positive. Finding the domain is a critical first step because it tells you where the function is defined and where you can actually see it on a graph. Without knowing the domain, you could make mistakes about the function’s behavior. Always remember to check what values you can legitimately plug into your logarithmic machine!

Decoding the Range of Logarithmic Functions

Now that we've nailed the domain, let's explore the range of logarithmic functions. Unlike the domain, the range of a standard logarithmic function, such as f(x) = logₐ(x), is usually quite straightforward. As x approaches zero from the right, the logarithmic function goes to negative infinity. As x increases without bound, the logarithmic function goes to positive infinity. That means that, in its most basic form, the range of f(x) = logₐ(x) is all real numbers, or (-∞, ∞). This holds true because you can take the logarithm of any positive number, and the output will always be a real number.

But what happens when you have transformations like f(x) = log(x + 3) - 2? Well, these transformations might shift the graph, but they don't change the fundamental nature of the range. For this specific function, f(x) = log(x + 3) - 2, the “-2” at the end represents a vertical shift. It moves the entire graph down by 2 units. However, this vertical shift doesn't change the possible y-values that the function can take. It’s simply shifting the location of the graph on the y-axis, not restricting the y values themselves. So, even with the vertical shift, the range remains (-∞, ∞). The logarithmic function, even with a vertical translation, still extends to both negative and positive infinity.

The key takeaway is that the range of logarithmic functions is often all real numbers. It doesn't matter if you have horizontal or vertical shifts; the graph will still extend infinitely upwards and downwards. This understanding is crucial because it helps you know what kind of output values to expect from the function.

Solving the Problem: Domain and Range of f(x) = log(x + 3) - 2

Alright, time to put everything together and solve the problem. We’ve been asked to find the domain and range of the function f(x) = log(x + 3) - 2. Let’s review what we have learned:

• Domain: We determined that the domain is (-3, ∞). This is because the argument of the logarithm, (x + 3), must be greater than zero. Solving the inequality x + 3 > 0 gives us x > -3. Therefore, the domain consists of all real numbers greater than -3.
• Range: We established that the range is (-∞, ∞). This is because logarithmic functions have a range of all real numbers, and the vertical shift of -2 does not change the range.

So, looking at the options provided, the correct answer is:

D. domain: (-3, ∞); range: (-∞, ∞)

Therefore, understanding the domain and the range are essential when working with any function, and this is especially true for logarithmic functions. By applying the knowledge that the argument of a logarithm must be positive and that the range of a standard logarithmic function is all real numbers, we were able to determine the correct domain and range for the given function. Keep practicing, and you'll become a domain and range expert in no time!