Finding domain and ranges for composition function

[Coco12]

Homework Statement

Find the domain and range of (f+g)(x)
Given that
F(x)= 1/(x-1) and g(x) = sqrt(x)

Homework Equations

The Attempt at a Solution

I added the two functions to get 1/(x-1) + sqrt(x)

To get domain, i look at the restrictive x values for both to get x greater than 0 and not equal to 1.
However how do i find the range?

 

[Mark44]

Homework Statement

Find the domain and range of (f+g)(x)
Given that
F(x)= 1/(x-1) and g(x) = sqrt(x)

Another question let's say:
f(x) = x+3
G(x)= -x-4
Find the domain and range of (f*g)(x)

Homework Equations

The Attempt at a Solution

I added the two functions to get 1/(x-1) + sqrt(x)To get domain, i look at the restrictive x values for both to get x greater than 0 and not equal to 1.
However how do i find the range?

Graph both functions separately, preferably using different colors for each graph. Then, using a third color, or a marker pen, go over the parts of the two graphs for which x ≥ 0 and x ≠ 1. From that graph it should be evident what the range of the sum function is.

2) I multiplied f(x) and g(x) together to get -x^2-7x-12
My domain I got all real numbers, again how do i get the range?

The graph of the product function is a parabola that opens downward. You should already have a technique for finding the vertex of a parabola. Let's say that the vertex is at (a, b). Then the range will be {y| -∞ < y ≤ b}.

BTW, you misposted this problem in one of the physics sections. This is a precalc type problem, so I moved it to this section.

 

[Coco12]

is there another way to find the range without graphing it?
the answer is y less than -0.7886 or y greater than 2.2287.
How can you get to that precise of an answer when graphing it by hand??

 

[Mark44]

They're probably using a graphing calculator or some graphing utility.

 

[Coco12]

For the second graph (f+g)(x)
The range in the back of the book says y less than .75 but u can see clearly in the graph that it is less than 1 ... What is going on?

 

[Coco12]

They're probably using a graphing calculator or some graphing utility.

I drew the graph out it is just a sketch and would not be accurate on the range.. Are you expected to know precisely where it is?

 

[Mark44]

View attachment 64817
For the second graph (f+g)(x)
The range in the back of the book says y less than .75 but u can see clearly in the graph that it is less than 1 ... What is going on?

The graphs don't show the sum or difference of the two functions. Those (sum or difference) are the graphs of interest here.

 

[Coco12]

The graphs don't show the sum or difference of the two functions. Those (sum or difference) are the graphs of interest here.

The thing is how do you graph 1/(x-1) + sqrt (x)?

 

[Mark44]

You can get a rough graph by doing it by hand. Pick values of x there are in the common domain for both functions (x ≥ 0, and x ≠ 1), the calculate the y-values.

As I said before, they are probably using a graphing calculator or computer graphics package.

 

[Coco12]

So for example I would take 0.5 and plug it into the composite formula?

 

[Coco12]

But the range won't be accurate right?