Finding H(x), Domain, And Range: A Step-by-Step Guide

Hey Plastik Magazine readers! Let's dive into some math, specifically focusing on how to determine a new function h(x) when you're given two other functions, f(x) and g(x). We'll also get into the important concepts of domain and range. Don't worry, it's not as scary as it sounds! This guide will break it down step by step, making it easy to understand. So, grab your coffee, and let's get started!

Understanding the Basics: Functions and Composition

First off, let's clarify what functions are. Think of a function like a machine. You put something in (an input, usually represented by x), and the machine does something to it, spitting out a new value (the output, often denoted as f(x) or g(x)). In our case, we have two machines: f(x) = x² and g(x) = √(4x - 5).

The function f(x) = x² is a simple one; it takes any input x and squares it. So, if you input 2, the output is 4 (because 2² = 4). The second function, g(x) = √(4x - 5), is a bit trickier because it involves a square root. Remember, you can only take the square root of a non-negative number. This detail is very important for determining the domain of g(x), as we'll see later. Finally, the composition of functions, like in our h(x), takes the output of one function and uses it as the input for another (or combines them in a defined way). When we define h(x) = f(x) * g(x), we're combining these two functions to create a new one. This essentially means we're going to multiply the outputs of f(x) and g(x) for a given x value. So, for the same x input, the machine of h(x) will apply f(x) and g(x), and then multiply the outputs of these two functions. This is where it gets interesting, so let's get our hands dirty and determine h(x).

Now that you know the functions and their properties, it's time to find the composition of the function. We're going to multiply the functions f(x) and g(x), to find our new function h(x).

Determining h(x): The Combination of f(x) and g(x)

Alright, let's figure out how to find h(x) when we know f(x) = x² and g(x) = √(4x - 5). Since h(x) = f(x) * g(x), we simply multiply the expressions for f(x) and g(x) together. So, that means: h(x) = x² * √(4x - 5). That's it, guys! We've found the formula for h(x). It's pretty straightforward, right? We just took the x² from f(x) and multiplied it by the square root expression from g(x).

However, although we found the explicit function, the game isn't over yet. The next step is to examine the domain and range of this new function h(x). Why are these details important? Because they tell us about the valid input and output values for our function. Knowing the domain ensures that we're only using valid input values (those for which the function is defined), and knowing the range tells us what output values are possible. Now, let's get to the fun part: finding the domain and range of h(x).

Finding the Domain of h(x): Permissible x Values

The domain of a function is the set of all possible input values (x-values) for which the function is defined. When dealing with h(x) = x² * √(4x - 5), we need to consider any restrictions on x that would make the function undefined. The key here is the square root. We know that the square root of a negative number is not a real number. Therefore, the expression inside the square root, which is (4x - 5), must be greater than or equal to zero.

Let's write this as an inequality: 4x - 5 ≥ 0.

Now, let's solve this inequality for x: Add 5 to both sides: 4x ≥ 5. Divide both sides by 4: x ≥ 5/4. So, the domain of h(x) is all x values that are greater than or equal to 5/4. In interval notation, this is written as [5/4, ∞). This means that you can only plug in values of x that are 5/4 or larger. If you try to plug in a value smaller than 5/4, you'll end up with a negative number inside the square root, and the function h(x) will not produce a real number result. This is why knowing the domain is so crucial – it tells us the valid inputs for our function.

Determining the Range of h(x): Possible Output Values

The range of a function is the set of all possible output values (y-values or h(x) values) that the function can produce. Finding the range can be trickier than finding the domain, but let's break it down for h(x) = x² * √(4x - 5). We know that the domain is x ≥ 5/4. When x = 5/4, the value inside the square root is zero, and the function equals 0. So, we know that h(5/4) = (5/4)² * √(4(5/4) - 5) = 0*. This is our minimum value for h(x).

As x increases from 5/4, both x² and √(4x - 5)* will increase. Since both factors are non-negative for x ≥ 5/4, their product, h(x), will also increase. This means that h(x) will never be negative, and it will increase without bound as x increases. So, the range of h(x) will be all non-negative real numbers, starting at 0 and going to infinity. Therefore, the range is [0, ∞). Note that the range is expressed in the same format as the domain. The domain and range give us a complete picture of the behavior of our function h(x). We know where it starts (at x=5/4, where h(x) is 0) and we know that it increases, without bound, to positive infinity. Pretty cool, right?

Conclusion: Wrapping It Up

So there you have it, folks! We've successfully determined h(x), which is x² * √(4x - 5), and we found its domain to be [5/4, ∞) and its range to be [0, ∞). We've explored how to combine functions and how to find the domain and range of the resulting function. Remember, the domain is all possible valid x values and the range is all possible output values. These concepts are fundamental in understanding and working with functions in mathematics.

Keep practicing, and you'll get the hang of it! Math can be fun, and with a step-by-step approach, it can be easily conquered. Until next time, keep exploring the fascinating world of mathematics! Bye guys! Hopefully, this was fun to learn. If you're looking for more guidance, check out our other articles!