Vertex, Domain, And Range Of F(x) = |x-5| + 10

by Andrew McMorgan 47 views

Hey guys! Today, we're diving into the absolute value function f(x) = |x-5| + 10. We'll figure out its vertex, domain, and range. Buckle up; it's gonna be a fun ride!

Understanding the Vertex

When we talk about the vertex of an absolute value function, we're essentially pinpointing its lowest (or highest) point. Think of it as the tip of the V that the function's graph forms. For a function in the form f(x) = a|x - h| + k, the vertex is simply the point (h, k). Identifying this point is super important because it tells us a lot about the function's behavior and how it sits on the coordinate plane.

In our case, f(x) = |x - 5| + 10, we can easily spot that h = 5 and k = 10. This means our vertex is at the point (5, 10). What does this tell us? Well, it means the lowest point of our V-shaped graph is at x = 5 and y = 10. This is where the absolute value function changes direction, and it's a crucial reference point for understanding the function's range and overall behavior. Remember, the vertex isn't just a point; it's the foundation upon which the rest of the graph is built.

Knowing the vertex helps us visualize and analyze the function more effectively. For instance, we know that the graph will be symmetrical around the vertical line x = 5, which passes through the vertex. This symmetry is a key characteristic of absolute value functions, and understanding it makes it easier to sketch the graph and predict the function's values for different inputs. So, the vertex (5, 10) is more than just a coordinate; it's a gateway to understanding the entire function. Isn't that neat?

Delving into the Domain

The domain of a function is all about figuring out what x values you can plug into the function without causing any mathematical mayhem. In simpler terms, it's the set of all possible inputs that the function can handle. For absolute value functions, like ours, the domain is usually pretty straightforward because there aren't many restrictions on what you can put inside an absolute value. You can throw in positive numbers, negative numbers, zero – basically anything you want!

For the function f(x) = |x - 5| + 10, there's no value of x that would make the function undefined. We don't have to worry about dividing by zero, taking the square root of a negative number, or any other common mathematical pitfalls. This means that x can be any real number. We can express this mathematically as x ∈ ℝ, which means x belongs to the set of all real numbers. Alternatively, we can write the domain in interval notation as (-∞, ∞). This notation tells us that the function is defined for all values from negative infinity to positive infinity.

So, whether you're dealing with a tiny fraction, a massive integer, or any number in between, you can always plug it into f(x) = |x - 5| + 10 without any issues. This is a great feature of absolute value functions because it makes them easy to work with in many different contexts. The domain of (-∞, ∞) simply means that our function is ready and willing to accept any input you throw its way. How cool is that?

Exploring the Range

Alright, let's talk about the range. The range of a function is all about figuring out what possible y values (or f(x) values) the function can spit out. It's the set of all possible outputs you can get from plugging in different x values. For absolute value functions, the range is closely tied to the vertex because the absolute value always returns a non-negative value.

In our function, f(x) = |x - 5| + 10, the absolute value part, |x - 5|, will always be greater than or equal to zero. No matter what x value you plug in, the absolute value will never be negative. This is because the absolute value of a number is its distance from zero, and distance can't be negative. So, the smallest value |x - 5| can be is 0, which happens when x = 5.

Now, let's think about what happens when we add 10 to |x - 5|. Since |x - 5| is always greater than or equal to 0, f(x) = |x - 5| + 10 will always be greater than or equal to 10. This means the smallest possible value for f(x) is 10, which occurs at the vertex (5, 10). As x moves away from 5 in either direction, the value of |x - 5| increases, and so does the value of f(x). There's no upper limit to how large f(x) can get, so it can go all the way to infinity.

Therefore, the range of f(x) = |x - 5| + 10 is all y values greater than or equal to 10. We can write this in set notation as {y | y ≥ 10}, which means "the set of all y such that y is greater than or equal to 10." In interval notation, we write the range as [10, ∞). The square bracket on the 10 indicates that 10 is included in the range, while the parenthesis on the infinity symbol indicates that infinity is not a specific number but rather an unbounded concept. So, the range tells us that our function's graph will never go below y = 10, and it will extend upwards indefinitely. Isn't it awesome how much information we can get from just looking at the function's equation?

Putting It All Together

So, to recap, for the function f(x) = |x - 5| + 10:

  • The vertex is at (5, 10).
  • The domain is (-∞, ∞), meaning x can be any real number.
  • The range is [10, ∞), meaning f(x) will always be greater than or equal to 10.

Given these findings, let's evaluate the options provided:

A. domain: (-∞, ∞); range: f(x) ≥ -5 (Incorrect. The range should be f(x) ≥ 10)

B. domain: (-∞, ∞); range: f(x) ≥ 10 (Correct!)

C. domain: x ≥ 5; range: f(x) ≥ Discussion category (Incorrect. The domain is all real numbers, and the range is f(x) ≥ 10)

So, the correct answer is B.

Wrapping up, absolute value functions might seem intimidating at first, but once you break them down and understand their key components like the vertex, domain, and range, they become much easier to handle. Keep practicing, and you'll become a pro in no time! Keep rocking it!