Rational function with an absolute value

[timCasper]

Hi, i need a quick help with my math homework. If it's possible, I would be thankful for more detailed solution, without using matrices, or using graphs only.


1. [MATH]f(x)=\frac{3|x|−3}{|x−1|}[/MATH], [MATH]x \neq 1[/MATH]. Find the set of values of this function.

2. [MATH]f(x)=\frac{2|x|−4}{|x−2|}[/MATH], [MATH]x \neq 2[/MATH]. Find the set of values of this function.

3. [MATH]f(x)=\frac{4−px}{x−p}[/MATH], [MATH]|p| \neq 2[/MATH]. Find all of [MATH]p[/MATH] values, for which the [MATH]f[/MATH] function is increasing in the range [MATH](p,+\infty)[/MATH].

4. [MATH]g(x)=\frac{mx+m+6}{x+m}, m \neq 3, m \neq 2[/MATH]. Find all of [MATH]m[/MATH] values, for which the [MATH]g[/MATH] function in decreasing in the range [MATH](−\infty,−m)[/MATH].

 

[HallsofIvy]

What do you understand about this problem and what have you tried? It's impossible to know what hints or suggestions would help you.

 

[Dr.Peterson]

Hi, i need a quick help with my math homework. If it's possible, I would be thankful for more detailed solution, without using matrices, or using graphs only.

1. [MATH]f(x)=\frac{3|x|−3}{|x−1|}[/MATH], [MATH]x \neq 1[/MATH]. Find the set of values of this function.

2. [MATH]f(x)=\frac{2|x|−4}{|x−2|}[/MATH], [MATH]x \neq 2[/MATH]. Find the set of values of this function.

3. [MATH]f(x)=\frac{4−px}{x−p}[/MATH], [MATH]|p| \neq 2[/MATH]. Find all of [MATH]p[/MATH] values, for which the [MATH]f[/MATH] function is increasing in the range [MATH](p,+\infty)[/MATH].

4. [MATH]g(x)=\frac{mx+m+6}{x+m}, m \neq 3, m \neq 2[/MATH]. Find all of [MATH]m[/MATH] values, for which the [MATH]g[/MATH] function in decreasing in the range [MATH](−\infty,−m)[/MATH].

I'd like to know the context of the question, and specifically why you would think matrices would apply, and why you don't want to use a graph. Were there any such instructions attached to the problem?

For #1, I tried some standard ways to find the range without graphing, and peculiar things happened. I reverted to just breaking the function into cases, and that led me into what amounts to graphing (by which I don't mean using a computer to graph it, but thinking about the graph!). Knowing what you are learning would help a lot.