106Q3. f(f(x)) = f(x)
This is clearly valid iff
=> f(x) = x
so 1 such function is possible?
1) Find the no. of solutions of logx3 = 5x-7.
2) If a line intersects the curve -
y= x4 +cx3 +12x2 -5x +2 in 4 distinct points, find the possible integral values of c.
3) Let a function f be defined as -
f : {1,2,3,4} → {1,2,3,4}.
If f satisfies - f(f(x)) = f(x), x ε {1,2,3,4},
then the no. of such functions is _____.
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7 Answers
106
21for 1st one
log_x 3 = \frac{1}{log_{3}x}=5x-7
\Rightarrow \frac{1}{5x-7}=log_3x
\frac{1}{5x-7} is a rectangular hyperbola....i suppose u can draw that
log_3 x can be easily drawn
so no of solns are no of points of intersection of rect hypebola 1/(5x-7) and log3x
which r two
66Q2) First observe that the linear and constant terms are irrelevant. Indeed, the graph of y=P(x) (where P(x) is a polynomial) meets the line y = m x +b in four distinct points provided the graph of y = P(x) + ax+d meets the line y = (m+a)x + (b+d) at four distinct points. Also a translation of the origin does not affect the condition: The graph of y = P(x) meets the line y =mx+b in four distinct points provided the graph of y =P(x-α) meets the straight line y = m(x-α) + b in four distinct points.
So, lets first transform the origin to the point x=h, y=0 so that the cubic term vanishes. For that we substitute x-h in place of x and equate the coefficient of x3 to zero. This gives h =c/4. And the resulting polynomial becomes (I neglect the linear and constant terms since they have no bearing in the problem)
x4 - 3(c2-32)8 x2
The graph of this polynomial which has a double root at =0 and opens upwards will be intersected by a straight line in 4 distinct points provided c2 > 32 which gives infinitely many integral values of c.
13Thankyou Anant sir n eragon bhai....
Waise, fr Q1 i did it the same way i.e. graphically, was looking fr an algebraic sol.n though.