The domain means the values of xx that make the equation untrue. So, we need to find the values that xx cannot equal.
For square root functions, xx cannot be a negative number. sqrt(-x)√−x would give us isqrt(x)i√x, where ii stands for imaginary number. We cannot represent ii on graphs or within our domains. So, xx must be larger than 00.
Can it equal 00 though? Well, let's change the square root to an exponential: sqrt0 = 0^(1/2)√0=012. Now we have the "Zero Power Rule", which means 00, raised to any power, equals one. Thus, sqrt0=1√0=1. Ad one is within our rule of "must be greater than 0"
So, xx can never bring the equation to take a square root of a negative number. So let's see what it would take to make the equation equal zero, and make that the edge of our domain!
To find the value of xx the makes the expression equal to zero, let's set the problem equal to 00 and solve for xx:
0= sqrt(7x+35)0=√7x+35
square both sides
0^2 = cancelcolor(black)(sqrt(7x+35)^cancel(2)
0=7x+35
subtract 35 on both sides
-35=7x
divide by 7 on both sides
-35/7 = x
-5 = x
So, if x equals -5, our expression becomes sqrt0. That is the limit of our domain. Any smaller numbers than -5 would give us a square root of a negative number.
