# Thread: Rational functions and their graphs,rational inequalities, transformations

1. ## Rational functions and their graphs,rational inequalities, transformations

Hi all,

I am an army wife taking a distance learning college algebra course. It is condensed- 8 weeks- yikes! And I am having some trouble with a few problems on my midterm review. If you can give me any assistance, I would greatly appreciate it!

1. Find the domain of (f/g)x when f(x)=6x ^2-3

and g(x)= x^2-2x-3

2. Find the domain of the composite function f with g

f(x)=2/x+7 g(x)=28/x

3. Given functions f and g, determine the domain of f+g
f(x)= 4x+5 g(x)=5/x+9

5. Graph the polynomial function x^4+ 4x^3 +4x^2
( you don't have to solve, can you just tell me how i go about graphing it?)

6. graph the rational function -4x/x+1

(dont have to solve, but can you give me steps to graphing it)

Thank you all so much for your help! This online course tells me the problems I got wrong on the midterm review, unfortunately It does not tell me the right answer or how to solve them correctly. God bless all of you math geniuses! I hope I can get there someday

2. Originally Posted by ashmo100388
Hi all,

I am an army wife taking a distance learning college algebra course. It is condensed- 8 weeks- yikes! And I am having some trouble with a few problems on my midterm review. If you can give me any assistance, I would greatly appreciate it!

1. Find the domain of (f/g)x when f(x)=6x ^2-3

and g(x)= x^2-2x-3

2. Find the domain of the composite function f with g

f(x)=2/x+7 g(x)=28/x

3. Given functions f and g, determine the domain of f+g
f(x)= 4x+5 g(x)=5/x+9

5. Graph the polynomial function x^4+ 4x^3 +4x^2
( you don't have to solve, can you just tell me how i go about graphing it?)

6. graph the rational function -4x/x+1

(dont have to solve, but can you give me steps to graphing it)

Thank you all so much for your help! This online course tells me the problems I got wrong on the midterm review, unfortunately It does not tell me the right answer or how to solve them correctly. God bless all of you math geniuses! I hope I can get there someday
Hi!

$\displaystyle f(x) = 6x^2 - 3$

$\displaystyle g(x) = x^2 - 2x - 3$

$\displaystyle (\frac{f}{g})(x) = \frac{6x^2 - 3}{x^2 - 2x - 3}$

The domain of a polynomial function is all real numbers. However, when it is in the form of a rational function, the denominator can not = 0. First factor the denominator.

$\displaystyle (\frac{f}{g})(x) = \frac{6x^2 - 3}{x^2 - 2x - 3}$

$\displaystyle = \frac{6x^2 - 3}{(x +1)(x-3)}$

So, the domain is all real numbers except at x = -1 and x = 3

So in interval notation: $\displaystyle (-\infty,-1)$ U $\displaystyle (-1,3)$ U $\displaystyle (3, \infty)$

Back in a sec with the next ones. just didn't want you to have to wait forever!

3. ## #2

f(g(x)) means that you will put g(x) into f(x) everywhere there is a x. Like this:

$\displaystyle f(x) = \frac{2}{x + 7}$

$\displaystyle g(x) = \frac{28}{x}$

$\displaystyle f(g(x)) = \frac{2}{(\frac{28}{x} + 7)}$

Get a common denominator:

$\displaystyle = \frac{2}{\frac{28 + 7x}{x}}$

Then multiply the top by the reciprocal of the bottom to get rid of the fraction in the denominator:

$\displaystyle = \frac{2}{1} \cdot \frac{x}{28 + 7x}$

$\displaystyle = \frac{2x}{28 + 7x}$

Now to find where the domain is restricted, put the denominator = 0

7x + 28 = 0

x = -4

So, the domain is all real numbers except when x = -4, or $\displaystyle (-\infty,-4)$ U $\displaystyle (-4,\infty)$

4. ## #3

$\displaystyle (f+g)(x)$ is the same as f(x) + g(x)

$\displaystyle f(x) = 4x + 5$

$\displaystyle g(x) = \frac{5}{x + 9}$

$\displaystyle (f + g) = (4x + 5) + (\frac{5}{x + 9})$

Get a common denominator:

$\displaystyle = \frac{4x (x + 9) + 5 (x + 9) + 5}{x + 9}$

Distribute and collect like terms:

$\displaystyle = \frac{4x^2 + 36x + 5x + 45 + 5}{x + 9}$

$\displaystyle = \frac{4x^2 + 41x + 50}{x + 9}$

Now you have a rational polynomial function. Like I showed you in the first ones, put the denominator = 0 and solve for x. This will show you where your domain in restricted. See if you can try it on this one.

5. Originally Posted by ashmo100388
[snip]
3. Given functions f and g, determine the domain of f+g
f(x)= 4x+5 g(x)=5/x+9

5. Graph the polynomial function x^4+ 4x^3 +4x^2
( you don't have to solve, can you just tell me how i go about graphing it?)

6. graph the rational function -4x/x+1

[snip]
3. Look at the domain of each function. The domain of f + g is what's common to both f and g ... In this case, all real numbers except x = -9.

5. Note that $\displaystyle x^4 + 4x^3 + 4x^2 = x^2 (x^2 + 4x + 4) = x^2 (x + 2)^2$.

6. Note that $\displaystyle \frac{-4x}{x+1} = -4 \left( \frac{x}{x + 1} \right) = -4 \left( \frac{(x + 1) - 1}{x + 1} \right) = -4 \left( 1 - \frac{1}{x + 1} \right) = \frac{4}{x + 1} - 4$.

I hope you recognise it as a rectangular hyperbola.