Understanding the Domain of Functions in Math

What is the domain of the function f(x) = 1/(x - 2)?

• A. All real numbers
• B. x ≠ 0
• C. x ≠ 1
• D. x ≠ 2

Answer: (D)

The domain of a function consists of all the values that the independent variable, in this case \( x \), can take without causing the function to be undefined. For the function \( f(x) = \frac{1}{x - 2} \), the denominator \( (x - 2) \) cannot equal zero since division by zero is undefined. To find where this occurs, set the denominator equal to zero: \[ x - 2 = 0 \] Solving this gives: \[ x = 2 \] Thus, the function \( f(x) \) is undefined when \( x = 2 \). Therefore, all real numbers are included in the domain except for \( x = 2 \). This leads to the conclusion that the domain of the function is all real numbers except for 2, which is expressed as \( x \neq 2 \). The other choices, while referring to other values or conditions, do not reflect the true restriction on the function's domain as accurately as the correct choice does. The correct answer clearly defines the specific value that causes the function to be undefined.

When it comes to functions in mathematics, understanding the domain is crucial. But what does that even mean? Let’s break it down in a way that makes sense, especially if you’re preparing for a college math placement test.

Imagine you’re going on a road trip with your friends. You’ve mapped out your route, but there’s a stretch of road that’s blocked. That’s a bit like what happens with functions: the domain is all about what values you can plug into a function before hitting a “roadblock”—or, in mathematical terms, making the function undefined.

Take, for instance, the function ( f(x) = \frac{1}{x - 2} ). Here’s where the fun begins! We need to identify which values of ( x ) keep this function on the road. The tricky part here is the denominator. For any function, we can't have division by zero, as it leads us to a dead end. So, let’s find out where that happens by setting the denominator equal to zero.

[

x - 2 = 0

]

When we solve this, we find that ( x = 2 ). That means when we input 2 into our function, we’ll hit a wall—it becomes undefined. So, what does this tell us about the domain? Pretty simply, it’s all real numbers except for 2. Mathematically, this is expressed as ( x \neq 2 ).

Now, let’s think about the other answer choices you might see on a math placement test. You might think that the domain could be defined by options like ( A. All real numbers ), ( B. x \neq 0 ), or ( C. x \neq 1 ). However, none of these correctly point out the critical value that causes the function to get stuck. Remember, while A includes all real numbers—which sounds good—it falsely encompasses 2; and the other two options mention values that don't affect our function at all.

This brings me to a crucial point—whenever you’re faced with these kinds of questions, always look for the specific values that cause issues for the function. It’s like a game of dodgeball; you need to know which balls (or values) to avoid to stay in the game. Keeping this in mind can help clarify many math concepts that may initially feel overwhelming.

To make sure you got this? Let’s bring it full circle. The domain of ( f(x) = \frac{1}{x - 2} ) captures all real numbers except the number 2. So next time you tackle similar problems, remember: it’s all about figuring out where the endpoints lie and dodging those pesky undefined spots.

So as you prep for that test—and perhaps a few nerves kick in—don’t fret. Just remember your math road trip rules, understand your domains, and you’ll be on your way to making solid decisions about which values are fair game and which ones you need to steer clear of!