The figure below shows the graph of a function $f(x)$. How many solutions does the equation $f(f(x)) = 15$ have?

Question

From the graph we can find that, $f(-5)=f(-1)=f(9)=0$

Answer 1

Hint:

If we restrict ourselves just on this graph, I mean we accept no other graph exists outer than $(-10,13)$. Then you need just to find the solution of $f(x)=4$, which is 4 distinct solutions $x_1,x_2,x_3,x_4$.

Answer 2

As Babak points out, we assume the domain of our function is restricted to the interval $(-10, 13)$:

• When $f(x) =4,\;$ $f(f(4)) = 15:\;$
  
  Find the x-values that correspond to all the points where $f(x) = 4$: you should find four such $x$-values: $x = -8, 1, 7.5, 10$.

• When $f(x) = 12$, note that $f(f(12)) = 15:$
  
  Find the x-values that correspond to where $f(x) = 12$. Again, there should be three or four such values, depending on whether you include the value at the far, far left of the graph: (approximations by "eying up" $x =? = -10, x = 3.5, 5.25, 11.5$

Answer 3

We are to find the solutions for f(f(x)) = 15.

From the graph, f(4) = 15 and f(12) = 15.

The required solutions will be those values of x for which f(x) = 4 and f(x) = 12.

From the graph, the value of function f(x) = 4 at four different values of x, namely x = –8, 1, 7.5 and 10.

The value of the function f(x) = 12 at three different points, namely x = 3.5, 5.25 and 11.5.

Hence, the given equation has 7 solutions.