Question

Derivative and inverse function help!?
Use the derivative to show why f(x)=definite integral 1/(4+e^t)dt, [0, ln x], with x>0 has an inverse function

Answer 1

if f'(x) > 0 for all x in (0,inf) then f is strictly increasing, thus, it has an inverse.

or

if f'(x) < 0 for all x in (0,inf) then f is strictly decreasing, thus, it also has an inverse

Answer 2

so after i find the derivative of this function, i look at the... range?

Answer 3

what did you get for derivative

Answer 4

i just realized i didn't put that the equation is 1/SQRT(4+e^t)

but my derivative would be \[-\frac{ e^t }{ 2(e^t+4)^{3/2} }\]

Answer 5

f'(x) = 1/[x sqrt(4 + x) ]

Answer 6

again, f'(x) > 0 for all x in (0,inf) so f has an inverse

Answer 7

f wasn't just 1/sqrt(4 + e^t)

it was
\[f(x) = \int\limits\limits_{0}^{\ln(x)} \frac{1}{\sqrt{4+e^t}} dt\]

Answer 8

okay, wow, way to go wolfram derivatives... so because no matter what positive number is used for x, f'(x) is positive. would graphing help see the inverse?

Answer 9

well, if you can find a way to graph f, then sure. But graphing is not always reliable

Answer 10

here is the graph of f
https://www.wolframalpha.com/input/?i=+-tanh%5E%28-1%29%28sqrt%28e%5E%28ln%28x%29%29%2B4%29%2F2%29+%2C+x+from+0+to+10

Answer 11

https://www.wolframalpha.com/input/?i=1%2F%28x%28sqrt%284%2Bx%29%29%29%2C+x+from+0+to+10

does this work as the graph of f'(x)?

Answer 12

that one is f'(x), not f(x)

Answer 13

but you can certainly use the graph of f'(x). As you can see, f'(x) is always positive on (0,inf)

Answer 14

yeah, i graphed f ' (x). but do i need to figure out the inverse function? or is that not necessary?

Answer 15

no, the question only asks for the proof that f(x) has an inverse.

Answer 16

right, okay. so it has an inverse because the function is continuously increasing for all the values of x>0 ?

Answer 17

yes

Answer 18

though I wouldn't say continuously increasing but rather strictly increasing.

Answer 19

okay, strictly it is. Thank you so much!