Answer:
(-infinity, 9/8]
Explanation:
Domain is defined as all the possible x-values of a function.
[tex]f(x) = 7\sqrt{9-8x} +4[/tex]
Lets test some negative numbers.
[tex]f(-5) = 7\sqrt{9-8(-5)} +4 => 7\sqrt{49} +4 => a real solution[/tex][tex]f(-1) = 7\sqrt{9-8(-1)} +4 => 7\sqrt{17} +4 => a real solution[/tex]
OK! That means all negative numbers should work!
What about 0?
[tex]f(0) = 7\sqrt{9-8(0)} +4 => 7\sqrt{9} +4 => a real solution[/tex]
0 works..
Now, onto positive numbers?
[tex]f(1) = 7\sqrt{9-8(1)} +4 => 7\sqrt{1} +4 => a real solution[/tex]
[tex]f(2) = 7\sqrt{9-8(2)} +4 => 7\sqrt{-7} +4[/tex] => imaginary
Check the last value that works. It can only be 9/8 because that's where the square root becomes 0, which is the last value before you head into the imaginary numbers..
[tex]f(9/8) = 7\sqrt{9-8(9/8)} +4 => 7\sqrt{0} +4=>4[/tex]
(-infinity, 9/8]
