Grades 11-12 Video Solutions 2021-video 2021 11-12/21

Video Transcription

Problem number twenty-one states, the figure shows the graph of a function mapping the closed interval from negative five to five to the real numbers.  How many distinct solutions does the equation f of f of x equals zero have?  So we want to know for what values of x, if we put them into f of x and get the output and put that into f of x, we would get zero.  So first let's ask ourselves, for what values of x, we'll call them a, does f of x equal zero? And this will be when our function crosses the x-axis or y equals zero.  So we get minus four, minus two, two, and four. So whenever we put any of these values into the function, we will get a zero out.  Now let's figure out what values of x does f of x equal a? And here we have all of the values plotted as y equals a, and we can just see where the lines intersect.  So we get the x values of about minus 4.75, minus 4.5, 3.25, minus 2.5, minus 1, 1, 4.5, and 4.75.  So we get 8 x values, and when we plug in any of these, for example 1, we will get out a number, in our case 2, that when plugged back into our function will yield zero, which  it does, as f of 2 is equal to zero. So our final answer is 8. So the question asked us, how many distinct solutions does the equation f of f of x equals zero have?  The answer is 8, letter e.

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