Question 538·Hard·Math

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The graph of $y=f(x)$ is shown.

Which choice gives the number of solutions to the equation $f(f(x))=2$ ?

A composite function feeds the inner output into the outer function, so a nested equation asks you to work backward twice. Read the graph at the outer target height to find possible inner outputs. Then check which of those outputs the graph can actually produce. Don’t multiply the branches automatically: a value below the graph’s minimum gives no solutions.

Hints

Hint 1
In a composite function, the output of the inner $f$ becomes the input of the outer $f$ . Work backward from the outer output: which inputs make $f$ give an output of $2$ ?
Hint 2
The range is the set of outputs a function can produce. An upward-opening parabola cannot produce an output below its vertex, its lowest point. Which possible inner output is too low?
Hint 3
For the remaining inner output, look across the graph at that height. A horizontal line can meet both sides of a parabola, and each meeting gives a different input.

Step-by-step

Work backward through the graph
Step 1
Separate the inner and outer functions
No Desmos needed. The given graph shows the relevant heights.

In the composite function $f(f(x))$ , the inner output $f(x)$ becomes the input to the outer $f$ . Call that inner output $u$ : $u=f(x)$ . Then $f(f(x))=2$ becomes $f(u)=2$ . First find which values of $u$ give the outer function an output of $2$ .
Step 2
Read the possible inner outputs
The line $y=2$ meets the graph at $(-6,2)$ and $(4,2)$ . A point $(a,b)$ means $f(a)=b$ , so $f(-6)=2$ and $f(4)=2$ . The two possible values of the inner output $u$ are $-6$ and $4$ .
Step 3
Check which inner outputs are possible
The vertex, the parabola’s lowest point, is $(-1,-4)$ . Since the graph opens upward, every output satisfies $f(x)\ge -4$ . The inner output must be a value the graph can reach. Because $-6<-4$ , $f(x)=-6$ has no solutions. Only $f(x)=4$ remains.
Step 4
Count the inputs for the remaining output
The line $y=4$ meets both sides of the parabola, so two different $x$ -values satisfy $f(x)=4$ . Each makes the outer function output $f(4)=2$ . So $f(f(x))=2$ has $2$ solutions. Choice C.

Question 538·Hard·Math

Strategy

Hints

Step-by-step

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Strategy

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