Question 1066644: If f(-4)=5, which of the following could not be the inverse of f(x)?
a)g(x)=x-9
b)g(x)=x^2-11
c)g(x)=x^2-5x-4
d)g(x)=-x^2+3x+6
Please help. My textbook doesn't have an example for a problem like this so I don't know where to even start.
Found 2 solutions by ikleyn, Edwin McCravy:
Answer by ikleyn(52802)   (Show Source):
You can put this solution on YOUR website! .
You need to check which of the listed functions has the output -4 when the input is 5.
If the function from the list HAS the output -4, then it potentially CAN BE the inverse to f(x).
It the function from the list HAS NO the output -4, it CAN NOT be the inverse to f(x). Then it is your lovely function.
Answer by Edwin McCravy(20060)  (Show Source):
You can put this solution on YOUR website!
She is forgetting that only one-to-one functions can HAVE or BE
inverse functions. These are functions which pass both the
horizontal and vertical line tests. No 2nd degree polynomial
functions are one-to-one, because they have u-shaped graphs
(parabolas) which do not pass the horizontal line test (unless
their domain is restricted.
So b), c), and d) are not one-to-one functions, and are immediately
eliminated. The only possibility is a). But let's see why.
f(-4)=5 means this:
When -4 is substituted for x in f(x), the result is 5.
Then in order for g(x) to be the inverse of f(x), then g(x)
must be a one-to-one function and we have the vice-versa of
the above. That is:
when 5 is substituted for x in the right side of g(x), the result
must be -4.
a) g(x)=x-9
g(5)=5-9 = -4
So (a) is the correct answer.
Notice that choices c) and d) might trick a student into thinking
they are correct because you also get -4 when you substitute x=5
c)g(5)=5^2-5(5)-4 = 25-25-4 = -4
d)g(5)=-5^2+3(5)+6 = -25+15+6 = -4
But they are not one-to-one functions because a u-shaped graph cannot
pass the horizontal line test.
Edwin
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