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
?
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
.