If g is decreasing, then if g(a) > g(b), then a<b
so, for #1, since g(3x^2−2x)≥g(3x−2),
3x^2-2x <= 3x-2
3x^2 - 5x + 2 <= 0
(3x-2)(x-1) <= 0
2/3 <= x <= 1
Do the others in like wise.
Solve the following inequalities, if it is known that function g is decreasing on its domain.
1. g(3x^2−2x)≥g(3x−2), Dg=ℝ
2. g(x^3−4x)
3. g(5−x^2)≤g(3x−5), Dg=(−∞,4)
4. g(x^2−3x)≥g(4x−12), Dg=(−∞,0]
3 answers
But what do we do if the domain isn't all real numbers like the 3rd and 4th example?
When the domain is constricted, you'll have all equations set to the domain. Allow me to explain.
For instance "g(5−x^2)≤g(3x−5), Dg=(−∞,4)" (assuming it is increasing) Will give us the equation 5−x^2≤3x−5. To make sure the domain is in check, you got to create two more equations. These equations, as per the domain, are 5−x^2<4 and 3x−5<4 cuz x<4. Put em in a system and you have the answer.
For instance "g(5−x^2)≤g(3x−5), Dg=(−∞,4)" (assuming it is increasing) Will give us the equation 5−x^2≤3x−5. To make sure the domain is in check, you got to create two more equations. These equations, as per the domain, are 5−x^2<4 and 3x−5<4 cuz x<4. Put em in a system and you have the answer.