I've been going through this community and I Find this really helpful. About me(I know I should be precise but ya), I'm just a highschool student who can't afford any coachings/schools. Self schooling being my only option I'm trying to teach myself mathematics from some torrented books. I am on functions and their graphs and stuck with one question
The functions f and g are defined for x ∈ R by f(x) =4x − 2x^2; g(x)= 5x + 3.
(i) Find the range of f. (ii) Find the value of the constant k for which the equation gf(x) = k has equal roots.
Now, I do understand the composition of functions but I just don't understand what they are asking in this case,
g(fx()=k would result in
20x-10x^2 +3=k
Now this is a quadratic equation of second degree which should have 2 roots/solutions, but that's the case if the right hand side was zero and not k. I have absolutely no idea how to tackle this question and what's being asked in part ii of the question. Anyhelp would be highly appreciated
Best Answer
Your equation could be rewritten as $$10x^2-20x+(k-3)=0.$$
Recall that the roots of a quadratic equation $ax^2+bx+c=0$ are given by $$\frac{-b\pm\sqrt{b^2-4ac}}{2a}.$$ So the two roots are equal when $b^2-4ac=0$. Can you apply this to your problem?