Question 518347
The domain is all the values that x can take in accordance with the rules of algebra.
.
You were given:
.
{{{(7x+9)/(x^2+10x+16)}}}
.
and were asked to find the domain. Whenever you see a fraction, you should first suspect that there might be a division by zero that is involved. The rules of algebra prohibit division by zero.
.
Notice that the second degree trinomial in the denominator can be factored. It factors as shown below:
.
{{{(7x+9)/((x+8)*(x+2))}}} 
.
If either of the two factors in the denominator equal zero, then the entire denominator will be zero because when anything is multiplied by zero the product equals zero. But remember, the denominator is not allowed to equal zero because division by zero is not permitted. That being the case, set each of the factors equal to zero and solve for x. The resulting values of x will not be permitted because they would cause a division by zero.
.
{{{x + 8 = 0}}}
.
Solve for x by subtracting 8 from both sides to get:
.
{{{x = -8}}}
.
Next:
.
{{{x + 2 = 0}}}
.
Solve for x by subtracting 2 from both sides to get:
.
{{{x = -2}}}
.
This means that x cannot equal -8 or -2. Other than that there are no limits on x. 
.
So the answer is that the domain for x is all values from -infinity to +infinity except for x = -8 and x = -2. Those two values are excluded from the domain.
.
Hope this helps. Keep in mind to suspect a division by zero whenever you see a problem that involves division (such as a fraction) with a variable in the divisor. There may be other things to watch out for, but in this problem, that was the one limit to the domain ... no division by zero.