Difference between revisions of "1980 AHSME Problems/Problem 14"
Oaktree1998 (talk | contribs) (Created page with "As <math>f(x)=cx/2x+3</math>, we can plug that into <math>f(f(x))</math> and simplify to get <math>c^2x/2cx+6x+9 = x</math> . However, we have a restriction on x such that if <ma...") |
Oaktree1998 (talk | contribs) |
||
Line 1: | Line 1: | ||
As <math>f(x)=cx/2x+3</math>, we can plug that into <math>f(f(x))</math> and simplify to get <math>c^2x/2cx+6x+9 = x</math> | As <math>f(x)=cx/2x+3</math>, we can plug that into <math>f(f(x))</math> and simplify to get <math>c^2x/2cx+6x+9 = x</math> | ||
− | . However, we have a restriction on x such that if <math>x=-3/2</math> we have an undefined function. We can use this to our advantage. Plugging that value for x into | + | . However, we have a restriction on x such that if <math>x=-3/2</math> we have an undefined function. We can use this to our advantage. Plugging that value for x into <math>c^2x/2cx+6x+9 = x</math> yields <math>c/2 = -3/2</math>, as the left hand side simplifies and the right hand side is simply the value we have chosen. This means that <math>c=-3</math>, which is answer choice '''A'''. |
+ | |||
+ | Alternatively, after simplifying the function to <math>c^2x/2cx+6x+9 = x</math>, multiply both sides by <math>2cx+6x+9</math> and divide by <math>x</math> to yield <math>c^2=2cx+6x+9</math>. This can be factored to <math>x(2c+6) + (3+c)(3-c) = 0</math>. This means that both <math>2c+6</math> and either one of <math>3+c</math> or <math>3-c</math> are equal to 0. <math>2c+6=0</math> yields <math>c=-3</math> and the other two yield <math>c=3,-3</math>. The clear solution is <math>c=-3</math>. |
Revision as of 19:13, 9 February 2013
As , we can plug that into
and simplify to get
. However, we have a restriction on x such that if
we have an undefined function. We can use this to our advantage. Plugging that value for x into
yields
, as the left hand side simplifies and the right hand side is simply the value we have chosen. This means that
, which is answer choice A.
Alternatively, after simplifying the function to , multiply both sides by
and divide by
to yield
. This can be factored to
. This means that both
and either one of
or
are equal to 0.
yields
and the other two yield
. The clear solution is
.