A portion of the graph of y = f(x) is shown in red below, where f(x) is a quadratic function. The distance between grid lines is 1 unit. What is the sum of all distinct numbers x such that f(f(f(x)))=-3 ?
First, we note that there are two points on the graph whose y-coordinates are -3. These are (-4,-3) and (0,-3). Therefore, if f(f(f(x)))=-3, then f(f(x)) equals -4 or 0. There are three points on the graph whose y-coordinates are -4 or 0. These are (-2,-4), (-6,0), and (2,0). Therefore, if f(f(x)) is -4 or 0, then f(x) equals -2, -6, or 2. There are four points on the graph whose y-coordinates are -2 or 2 (and none whose y-coordinate is -6). The x-coordinates of these points are not integers, but we can use the symmetry of the graph (with respect to the vertical line x=-2) to deduce that if these points are (x1,-2), (x1,-2), (x3,2), and (x4,2), then x1+x2=-4 and x3+x4=-4. Therefore, the sum of all four x-coordinates is -8.
+118667 Mmm
A portion of the graph of y = f(x) is shown in red below, where f(x) is a quadratic function. The distance between grid lines is 1 unit. What is the sum of all distinct numbers x such that f(f(f(x)))=-3 ?
f(0) = -3 and f(-4)= -3
So now I need f(x)=0 and f(x)=-4
f(2)=0, f(-6)=0 f(-2)=-4
So now I need
f(x)=2, f(x)=-6 and f(x)=-2
f(2.8)=2, f(-6.8)=2 and none, and f(0.8)=-2 f(-4.8)=-2
f(2.8)=2
f(f(2.8)=f(2)=0
f(f(f(2.8))=f(f(2))=f(0)=-3
same for the others
so
the possible x values are 2.8, -6.8, 0.8 and -4.8 (all approximate)
sum = -8
Guest, the content of your answer may well be better than mine but it is so blocked together that it would be very difficult for people to decipher.
Presentation is important.
+129850 f(f(f(x)) = - 3
Take the inverse of both sides
f-1 f(f(f(x)) = f-1 (-3) = - 4 and 0
f(f(x) = f-1 (-3) = - 4 and 0
Take the inverse of both sides again
f-1 f(f(x) ) = f-1(-4) and f-1 f(f(x)) = f-1(0)
f(x) = -2 and f(x) = -6 and f(x) = 2
Take the inverse one last time
f-1f(x) = f-1(-2) and f-1 f(x) = f-1(-6) and f-1 f(x) = f-1(2)
x = f-1(-2) and x = f-1(-6) and x = f-1 (2)
Since x = - 2 is a line of symmetry
Then f-1(-2) can be written as -2 - a, -2 + a
And f-1(2) = can be written as -2 - b, -2 + b
And f-1(-6) = not on graph
So....the sum of the coordinates =
(-2 - a) + (-2 + a) + (-2 - b) + (-2 + b) = -8
