"Challenging" Algebra?

Question

I just wanted to check my answer to anyone who's willing to answer it... (PS an explanation would be extremely helpful!!)

Identify the domain of the function give by g(x) = [Square Root of (x+1)] all divided by (x^2 +4x +5) .
What if anything can you deduce about the range?


Identify the domain of the function give by g(x) = (x+1) all divided by (x^2 +4x -5) .
What if anything can you deduce about the range?

Answer 1

1)
g(x) = sqrt (x + 1) / (x^2 + 4x + 5)

Domain: can't have a negative under the square root, and can't have a zero in the denominator, so...
x + 1 >= 0
x >= -1

x^2 + 4x + 5 = 0
can't equal zero (no real roots)

So, the domain is x >= -1

Range... The top number (sqrt (x+1)) is always positive. And, the bottom number will always be positive because x^2 will be positive, and 4x+5 will always be positive (because the only negative numbers you can put in are between 0 and -1, which would result in 4x + 5 being positive.)
So, the range will always be greater than or equal to zero.

2)
g(x) = (x + 1) / (x^2 + 4x - 5)

Domain:
no zero in denominator, so
x^2 + 4x - 5 = 0
(x + 5)(x - 1) = 0
x can't equal -5 or 1

Range:
could be anything.

Answer 2

Hello

So we have [Square Root of (x+1)] / (x^2 +4x +5)

The Domain is all the "x" values that can be plugged - resulting in a y value.

Since we know its a square root "function" we know that the left most point of the graph is -1. If we plug anything smaller than -1 into the equation - we will get "no solution" since it will leave us with a negative under the square root.

So the Domain is all REALS x greater than or equal to -1.

The range is all possible y values.

Graph this and use the "max" key and you see that the highest point will be .21796. Also, as you place larger values of x into the equation you see that the y will get smaller and small. If you place -1 into the equation - we get y=0.

Thus the range is: .21796 >=y >= 0.

B) So we have (x+1)/ (x^2 +4x +5).

We can use any x value here since there is no square root.
Domain is all reals.

Also the highest point of the range is .207 and lowest is -1.2.

So the range is all y values greater than or equal to -1.2 and less than or equal to .207.

Hope this helps

Answer 3

g(x) = (x+1)^.5/(x^2+4x +5)
= (x+1)^.5/[(x+1)(x+4)]
The denominator becomes 0 when x = -1 or -4
So Domain is all real nunbers except -1 and -4
The numerator goes negative when x< -1 so the the function becomes imaginary if x< -1
Hence the domain is x >-1.

g(x) = (x+1)/[(x+5)(x-1)]
Denominator goes to 0 when x = -5 or +1
Domain is all real numbers except -5 and 1.

Answer 4

1)
domain : x>-1 ( sqrt not defined for argument < 0 )
and x not such that x^2 +4x +5 equals zero

2) x not such that x^2 +4x -5 = 0

Answer 5

I find it easier to answer these sort of questions by looking at the graphs, which see
http://i233.photobucket.com/albums/ee319/fjfk/twograph.jpg

The red line is the first equation with the radical.