Question 438·Medium·Math

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

Which choice gives a value of $k$ for which the equation $f(x)=k$ has exactly three real solutions?

For an equation like $f(x)=k$ , each intersection, or meeting point, of the graph and the horizontal line $y=k$ gives one real solution. If a curve rises, falls, then rises again, look at the heights where it turns. A line can meet all three stretches when its height is strictly between those turning-point heights. Touching a turn counts only once.

Hints

Hint 1
The equation $f(x)=k$ asks which inputs give the same output $k$ . Picture a horizontal line at height $k$ : each place it meets the curve gives one solution.
Hint 2
A turning point is where the curve changes from rising to falling or from falling to rising. Read the heights, not the $x$ -coordinates, of the two marked turning points. Where must a line sit to meet all three stretches?

Step-by-step

Count intersections on the given graph
Step 1
Translate solutions into intersections
No Desmos needed. The given graph already shows the curve and its turning points. A real solution to $f(x)=k$ is an input $x$ whose output is $k$ . So picture the horizontal line $y=k$ : each distinct place it meets the curve gives one solution.
Step 2
Find the heights that give three meetings
The marked turning points, where the curve changes direction, have heights $4$ and $-2$ . A horizontal line meets all three stretches only when its height is strictly between those two heights. So neither endpoint works: at an endpoint, the line touches a turn and meets the curve at only one other place. The needed range is $-2<k<4$ .
Step 3
Choose a height in the range
Of the choices, $k=0$ lies between $-2$ and $4$ . The line $y=0$ is the $x$ -axis, and the curve crosses it three times. So $f(x)=0$ has exactly three real solutions. Choice C.

Question 438·Medium·Math

Strategy

Hints

Step-by-step

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Strategy

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Step-by-step

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