Grades 11-12 Video Solutions 2013-Levels 11&12 Video Solutions 2013 problem17

Video Transcription

Question number 17, Vlad has drawn the graph of a function f mapping the real numbers to the real numbers,  composed of two rays and a line segment as shown in the diagram.  How many solutions does the equation f composed with f composed with f of x equal to 0 have?  So to begin, let's write down the equations  that determine these line segments.  For example, the line here passing through negative 4 will also pass through positive 4 on the y-axis, and so the slope is 1, and we can say  that the equation is y is equal to x plus 4.  This line is parallel to the other line with slope 1,  and that line would have the equation y is equal to x as it passes through the origin. And finally, the segment here lies on the line y is equal to negative x,  as we can compute the slope and we see it passes through the origin also.  Now, looking at our equation, f composed with f composed with f of x is equal to 0, provided that we evaluate the function f at negative 4,  which is an x-intercept, or 0, which is another x-intercept.  So this happens, provided that f composed with f of x is equal to negative 4, or 0, and then we have to go through both cases.  Now, f composed with f of x is equal to negative 4,  somewhere along the line y is equal to x plus 4.  And so we can quickly solve that, and that happens, provided that f of x is equal to negative 8, and f of x composed with f of x is equal to 0, provided that f of x is equal  to, again, negative 4, or 0, and we have more cases to consider.  Finally, f of x is equal to negative 8,  provided that x is equal to negative 12, as that solution will lie, again, on the line with equation y is equal to x plus 4.  And f of x will be equal to negative 4,  if and only if x is equal to negative 8, as we have already calculated, and that leaves us with checking when f of x will be equal to 0,  which we have also done.  We know that negative 4, or 0, are the solutions, and so we have finally four possibilities.  When x is equal to 0, negative 4, negative 8, negative 12,  f composed with itself twice will be equal to 0.  So the answer to number 17 is A.

Video Summary

The video discusses finding the number of solutions to the equation \( f(f(f(x))) = 0 \) for a piecewise function composed of two rays and a line segment. It details the equations of the function's components: two lines \( y = x + 4 \) and \( y = -x \). By evaluating \( f(f(x)) = -4 \) or \( 0 \) for these lines, the solutions are determined. The possible values for \( x \) are worked out as \( 0, -4, -8, \) and \( -12 \). Thus, there are four solutions to the equation, and the answer is option A.

Keywords

piecewise function

equation solutions

rays and line segment

nested function

mathematical analysis