SOLUTION: Find the domain and range of the following d) {{{y=7/(x+9)}}} e) {{{y=3/(x^2-4)}}}

Algebra ->  Functions -> SOLUTION: Find the domain and range of the following d) {{{y=7/(x+9)}}} e) {{{y=3/(x^2-4)}}}       Log On


   

Question 128507: Find the domain and range of the following

d)
y=7%2F%28x%2B9%29

e)
y=3%2F%28x%5E2-4%29



Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
# 1

d)

y=%287%29%2F%28x%2B9%29 Start with the given function


x%2B9=0 Set the denominator equal to zero. Remember, dividing by 0 is undefined. So if we find values of x that make the denominator zero, then we must exclude them from the domain.



x=0-9Subtract 9 from both sides


x=-9 Combine like terms on the right side





Since x=-9 makes the denominator equal to zero, this means we must exclude x=-9 from our domain

So our domain is:

which in plain English reads: x is the set of all real numbers except x%3C%3E-9

So our domain looks like this in interval notation


note: remember, the parenthesis excludes -9 from the domain

---------------------------------------------------------




Now to find the range, simply graph the function to get


note: ignore the vertical lines, they are not part of the graph and are asymptotes
graph%28500%2C500%2C-15%2C5%2C-10%2C10%2C%287%29%2F%28x%2B9%29%29


Looking at the graph, we can see that y can be any number except 0. So y%3C%3E0.




So our range is:

which in plain English reads: y is the set of all real numbers except y%3C%3E0

So our range looks like this in interval notation


note: remember, the parenthesis excludes 0 from the range





e)





y=%283%29%2F%28x%5E2-4%29 Start with the given function


x%5E2-4=0 Set the denominator equal to zero. Remember, dividing by 0 is undefined. So if we find values of x that make the denominator zero, then we must exclude them from the domain.




%28x-2%29%28x%2B2%29=0 Factor the left side (note: if you need help with factoring, check out this solver)




Now set each factor equal to zero:

x-2=0 or x%2B2=0

x=2 or x=-2 Now solve for x in each case


So our solutions are x=2 or x=-2



Since x=-2 and x=2 make the denominator equal to zero, this means we must exclude x=-2 and x=2 from our domain

So our domain is:

which in plain English reads: x is the set of all real numbers except x%3C%3E-2 or x%3C%3E2

So our domain looks like this in interval notation


note: remember, the parenthesis excludes -2 and 2 from the domain



---------------------------------------------------------




Now to find the range, simply graph the function to get




note: ignore the vertical lines, they are not part of the graph and are asymptotes

graph%28500%2C500%2C-15%2C5%2C-10%2C10%2C%283%29%2F%28x%5E2-4%29%29


Looking at the graph, we can see that y can be any number except 0. So y%3C%3E0.




So our range is:

which in plain English reads: y is the set of all real numbers except y%3C%3E0

So our range looks like this in interval notation


note: remember, the parenthesis excludes 0 from the range