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're going to pinpoint its vertex, figure out its domain, and nail down its range. Buckle up, it's gonna be a fun ride!

Finding the Vertex

First things first, let's talk about the vertex. For an absolute value function in the form f(x) = a|x - h| + k, the vertex is simply the point (h, k). This point represents either the minimum or maximum value of the function, depending on whether 'a' is positive or negative. Absolute value functions always produce a V-shape, and the vertex is that sharp turning point at the bottom (or top, if it's upside down!).

Now, in our case, we have f(x) = |x - 5| + 10. Comparing this to the general form, we can easily see that h = 5 and k = 10. Therefore, the vertex of our function is at the point (5, 10). This is the lowest point on the graph of the function, because the absolute value part, |x - 5|, will always be greater than or equal to zero. Adding 10 to it just shifts the whole graph up by 10 units. To confirm this, you could plot a few points around x = 5, like x = 4 and x = 6, and see that the corresponding y-values are greater than 10. Understanding how transformations affect the basic absolute value function y = |x| is key here. Remember, subtracting a number inside the absolute value (like the -5) shifts the graph to the right, and adding a number outside the absolute value (like the +10) shifts the graph up.

So, the vertex (5, 10) is a crucial point. It tells us a lot about the behavior of the function. It's the starting point for understanding where the function is increasing and decreasing, and it plays a huge role in determining the range of the function, which we'll get to in just a bit. Visualizing the graph can be super helpful too! If you sketch it out, you'll see the V-shape with its point firmly planted at (5, 10).

Determining the Domain

Alright, let's move on to the domain of our function. The domain is all about the possible input values, or the 'x' values, that we can plug into the function without causing any mathematical mayhem. Think of it as what values of 'x' are allowed to play in the function's sandbox.

For the absolute value function f(x) = |x - 5| + 10, there are no restrictions on the values of 'x'. We can plug in any real number we want! Whether it's a positive number, a negative number, zero, a fraction, or even a crazy irrational number, the function will happily chug along and produce a valid output. This is because the absolute value of any real number is always defined. There's no division by zero to worry about, no square roots of negative numbers causing trouble, nothing that can break the function.

Therefore, the domain of f(x) = |x - 5| + 10 is all real numbers. We can express this in interval notation as (-∞, ∞). This means that the function is defined for all x-values from negative infinity to positive infinity. Knowing the common domains of basic functions like polynomials, rational functions, and radical functions can help you quickly determine the domain of more complex functions. Absolute value functions are similar to polynomials in that their domain is always all real numbers.

In simpler terms, you can substitute any number for 'x' in the equation f(x) = |x - 5| + 10, and you will always get a real number back as the result. This is a key characteristic of absolute value functions, and it makes finding the domain pretty straightforward.

Figuring Out the Range

Now, let's tackle the range. The range is all about the possible output values, or the 'f(x)' values (which are the same as 'y' values), that the function can produce. It's like asking, "What are all the possible heights the function can reach?"

Since we have an absolute value function, we know that the absolute value part, |x - 5|, will always be greater than or equal to zero. It can never be negative. The smallest possible value for |x - 5| is 0, which occurs when x = 5. This is important because it directly impacts the minimum value of the entire function.

Because |x - 5| is always greater than or equal to zero, the smallest possible value for f(x) = |x - 5| + 10 is when |x - 5| = 0. In that case, f(x) = 0 + 10 = 10. So, the function will never dip below 10. It can be 10, and it can be any number larger than 10, but it can't be anything smaller. Understanding the behavior of the absolute value function, particularly its non-negativity, is crucial for determining the range. Think about it: no matter what value you plug in for 'x', the absolute value will always make it positive (or zero), and then you're adding 10 to that.

Therefore, the range of f(x) = |x - 5| + 10 is f(x) ≥ 10. In interval notation, we write this as [10, ∞). This means that the function's output values start at 10 (inclusive) and go all the way up to positive infinity. The square bracket on the 10 indicates that 10 is included in the range.

The Answer

So, putting it all together:

  • Vertex: (5, 10)
  • Domain: (-∞, ∞)
  • Range: f(x) ≥ 10 or [10, ∞)

That means the correct answer is B. domain: (-∞, ∞); range: f(x) ≥ 10

Hope this helps you nail those absolute value function problems! Keep practicing, and you'll become a pro in no time!