Question

let f(x)=x^2-16. find f^1(x)

Answer 1

is the equation like this:
\[f(x)=x ^{2}-16.\]
For the second on is it
\[f ^{1}(x) \]
or
\[f ^{-1}\]
or\[f(1)\]?

Answer 2

I think she means the derivative of f(x).

Answer 3

yes the equation is like that and for the second part its the first one

Answer 4

If it is the derivative then se the product rule on the x^2 part. The derivative of a constant is zero.

Answer 5

if so, then
\[\prime= (2)x ^{2-1}\]
16 is constant.. Derivative of a constant is 0
for x^2 is will be \[n \times ax ^{n-1}\]
\[f \prime = 2x\]

Answer 6

these are the options cuz none of the ones you guys are saying make sence
\[4\sqrt{x}\]
\[\sqrt{x+16}\]
\[\frac{ x^2 }{ 16 }\]
\[\frac{ 1 }{ x^2-16}\]

Answer 7

are you looking for the inverse?

Answer 8

im guessing so thoses are the options and im confused

Answer 9

if you're looking for the inverse, then,
1.Switch x and y. We know y=f(x),
\[y=x^{2}-16\]
\[x=y ^{2}-16\]

Answer 10

2.solve for y
\[y ^{2}= x+16\]
\[y=\sqrt{x+16}\]
Therefore,\[\sqrt{x+16}\]

Answer 11

Did you get it?

Answer 12

oh yes i get it i was solving for y wrong

Answer 13

unfortunately this is an abuse of notation, because in this example
\[f^{-1}(x)\] is not a function

Answer 14

on the second step, you take the square root of both sides, to remove the square in y

Answer 15

thank you :)

Answer 16

you're welcome

Answer 17

the very last step should be
\[y^2=x+16\] so \[y=\pm\sqrt{x+16}\]

Answer 18

make sure to include the \(\pm\) in your answer

Answer 19

@satellite how will I know when to use \[\pm \sqrt{x}\]

Answer 20

if you have
\[a^2=b\] and you are solving for \(a\) it is \(a=\pm\sqrt{b}\)

Answer 21

for example if \(x^2=9\) then \(x=\pm\sqrt9\) or \(x=3,-3\)

Answer 22

Does it always have to be like that?

Answer 23

this is why this is a terrible problem
you are asked to find the inverse of a function, but the function is not one to one, so the inverse is not a function
written by a lousy math teacher

Answer 24

but in some books they will consider + as a separate function and - as a separate function

Answer 25

it would make sense for example if it said
"let \(f(x)=x^2-16\) for \(x>4\) then you could find the inverse