The graph of the function f , defined by , is shown in the xy -plane above. If the function

Question

The graph of the function f , defined by , is shown in the xy -plane above. If the function g (not shown) is defined by , what is one possible value of a such that ?

Accepted Answer

The correct answer is either 2 or 8. Substituting in the definitions for f and g gives and , respectively. If , then . Subtracting 10 from both sides of this equation gives . Multiplying both sides by gives . Expanding gives . Combining the like terms on one side of the equation gives . One way to solve this equation is to factor by identifying two numbers with a sum of and a product of 16. These numbers are and , so the quadratic equation can be factored as . Therefore, the possible values of a are either 2 or 8. Note that 2 and 8 are examples of ways to enter a correct answer. Alternate approach: Graphically, the condition implies the graphs of the functions and intersect at . The graph is given, and the graph of may be sketched as a line with y -intercept 10 and a slope of (taking care to note the different scales on each axis). These two graphs intersect at and .

Question the-gr