Dividing Functions And Finding Domains F(x) And G(x)

Hey guys! Today, we're diving into the fascinating world of functions, specifically focusing on how to divide them and figure out their domains. We'll be working with two functions, f(x) and g(x), and walking through the process step-by-step. So, buckle up and let's get started!

Defining Our Functions: f(x) and g(x)

Before we jump into the division, let's clearly define our functions. We have:

• f(x) = 5 / (x + 3)
• g(x) = 9 / x

These are rational functions, meaning they are fractions where the numerator and denominator are polynomials. Understanding this is crucial because it impacts how we deal with domains later on.

Finding (f/g)(x): Dividing the Functions

So, the main question is: what happens when we divide f(x) by g(x)? This is represented as (f/g)(x), which is simply f(x) divided by g(x). Let's break it down:

(f/g)(x) = f(x) / g(x)

Now, we substitute the actual functions:

(f/g)(x) = [5 / (x + 3)] / [9 / x]

Dividing by a fraction is the same as multiplying by its reciprocal. So, we flip the second fraction and multiply:

(f/g)(x) = [5 / (x + 3)] * [x / 9]

Now, multiply the numerators and the denominators:

(f/g)(x) = (5 * x) / [(x + 3) * 9]

(f/g)(x) = 5x / [9(x + 3)]

(f/g)(x) = 5x / (9x + 27)

And there you have it! We've successfully divided f(x) by g(x). The resulting function, (f/g)(x), is 5x / (9x + 27). But our journey isn't over yet. We need to consider the domain of this new function.

Deciphering the Domain: Where is (f/g)(x) Valid?

The domain of a function is the set of all possible input values (x-values) for which the function produces a valid output. For rational functions like ours, the main thing we need to watch out for is division by zero. We can't have a zero in the denominator, as that would make the function undefined.

So, to find the domain, we need to identify any x-values that would make the denominator of (f/g)(x) equal to zero. Let's look at our simplified function:

(f/g)(x) = 5x / (9x + 27)

The denominator is (9x + 27). We need to find when this equals zero:

9x + 27 = 0

Subtract 27 from both sides:

9x = -27

Divide both sides by 9:

x = -3

This tells us that x = -3 makes the denominator zero, so it's not in the domain of (f/g)(x). But there's another crucial aspect to consider when determining the domain of (f/g)(x): the domains of the original functions, f(x) and g(x). Remember, we started with f(x) / g(x). If either f(x) or g(x) is undefined for a particular x-value, then (f/g)(x) will also be undefined at that x-value.

Let's look back at our original functions:

• f(x) = 5 / (x + 3). The denominator is (x + 3). If we set this equal to zero, we get x = -3. So, f(x) is undefined at x = -3.
• g(x) = 9 / x. The denominator is x. If we set this equal to zero, we get x = 0. So, g(x) is undefined at x = 0.

Also, we must consider when g(x) = 0, because we are dividing by g(x). However, in this case g(x) = 9/x never equals 0, so we don't need to worry about that.

Therefore, we have three critical values to exclude from the domain of (f/g)(x): x = -3 (from f(x)), x = 0 (from g(x)), and x = -3 (from the simplified (f/g)(x)). Notice that x = -3 appears twice, but we still only need to exclude it once.

Expressing the Domain in Interval Notation

Now that we've identified the values to exclude, let's express the domain using interval notation. We're excluding -3 and 0, so we'll have three intervals:

• From negative infinity up to -3: (-∞, -3). We use a parenthesis because -3 is not included.
• From -3 to 0: (-3, 0). Again, we use parentheses because neither -3 nor 0 are included.
• From 0 to positive infinity: (0, ∞). We use a parenthesis because 0 is not included.

To represent the entire domain, we combine these intervals using the union symbol (∪):

Domain of (f/g)(x): (-∞, -3) ∪ (-3, 0) ∪ (0, ∞)

This interval notation tells us that (f/g)(x) is defined for all real numbers except -3 and 0. And that, my friends, is the complete domain of our divided function!

Wrapping Up: Key Takeaways

Let's quickly recap what we've done:

1. We defined two functions, f(x) and g(x).
2. We found (f/g)(x) by dividing f(x) by g(x) and simplifying the result.
3. We determined the domain of (f/g)(x) by considering the restrictions from both the original functions and the simplified result.
4. We expressed the domain in interval notation.

Remember, when dividing functions, always pay close attention to the denominators and the domains of the original functions. By doing so, you can confidently navigate the world of function operations! This comprehensive approach ensures we don't miss any critical points when defining the domain.

Find f/g and its domain in interval notation, given f(x) = 5/(x+3) and g(x) = 9/x. Simplify the answer.

Dividing Functions and Finding Domains f(x) g(x) Explained