Find f/g (x), simplify and find its domain

[FritoTaco]

Hello,

I want to get this answer right but probably forgot a few math concepts.

f(x) = [(x - 4) / (x - 5)], g(x) = [(2(x - 4)] / (x - 6)

They want me to divide f/g(x).

My work,

[(x - 4) / (x - 5)] * [(x - 6) / 2(x - 4)] I used reciprocal and brought it on top with [(x - 4 /x - 5)]

I feel like I shouldn't have done that, could I have just canceled out x-4 from both equations instead without using reciprocal?

New work,

(x-5) / (2/x-6)

Am I now suppose to use reciprocal after? Also, I've been researching for 2 hours how to use LaTeX and I couldn't figure it out. I tried it on the forum page to use the $$ sign in front and end of the code but it just prints exactly how I had it when I click preview post. I know a lot of people use LaTeX, can someone help point me in right direction so my work can be shown neatly and easier on the eyes. I looked at the sticky thread but it didn't help me too much.

 

[ksdhart2]

I believe what you did is most likely the simplest method. You correctly identified that f(x) / g(x) = f(x) * [1/g(x)], so all that remains is putting it into practice. Perhaps seeing it in a more visual form will help you understand what is going on. By the way, I'm assuming that you missed a few parentheses, and meant to say that f(x) = (x-4)/(x-5).

\(\displaystyle \displaystyle \frac{x-4}{x-5}\cdot \frac{1}{\frac{2\left(x-4\right)}{x-6}}=\frac{x-4}{x-5}\cdot \frac{x-6}{2\left(x-4\right)}=\frac{\left(x-4\right)\left(x-6\right)}{\left(x-5\right)2\left(x-4\right)}\)

So, yes, you're correct that you can cancel an (x-4) term. Then, I believe that's as simplified as it can get.

As for LaTeX, what I usually do is use an external website called Symbolab Calculus Calculator. There, I can write out the math problem and copy and paste it as TeX code. Then you just put the code inside Tex tags. For instance, if the code you copied was for (x-4)/(x-5), you'd have this:

[tex]\displaystyle \frac{x-4}{x-5}[/tex]

The "\displaystyle" at the beginning is not included in the copy-and-paste from Symbolab, but I always include it whenever I use fractions, to make them look bigger. The difference is this:

Without: \(\displaystyle \frac{x-4}{x-5}\) With: \(\displaystyle \displaystyle \frac{x-4}{x-5}\)

 

[FritoTaco]

I believe what you did is most likely the simplest method. You correctly identified that f(x) / g(x) = f(x) * [1/g(x)], so all that remains is putting it into practice. Perhaps seeing it in a more visual form will help you understand what is going on. By the way, I'm assuming that you missed a few parentheses, and meant to say that f(x) = (x-4)/(x-5).

\(\displaystyle \displaystyle \frac{x-4}{x-5}\cdot \frac{1}{\frac{2\left(x-4\right)}{x-6}}=\frac{x-4}{x-5}\cdot \frac{x-6}{2\left(x-4\right)}=\frac{\left(x-4\right)\left(x-6\right)}{\left(x-5\right)2\left(x-4\right)}\)

So, yes, you're correct that you can cancel an (x-4) term. Then, I believe that's as simplified as it can get.

As for LaTeX, what I usually do is use an external website called Symbolab Calculus Calculator. There, I can write out the math problem and copy and paste it as TeX code. Then you just put the code inside Tex tags. For instance, if the code you copied was for (x-4)/(x-5), you'd have this:

[tex]\displaystyle \frac{x-4}{x-5}[/tex]

The "\displaystyle" at the beginning is not included in the copy-and-paste from Symbolab, but I always include it whenever I use fractions, to make them look bigger. The difference is this:

Without: \(\displaystyle \frac{x-4}{x-5}\) With: \(\displaystyle \displaystyle \frac{x-4}{x-5}\)

Wow, thank you so much for your help. The math problem and LaTeX.