Question

Please help. WILL MEDAL! Inverse functions @Hero @sleepyjess @jim_thompson5910 @robtobey @Erak

Answer 1

Determine if \[f(x) = \frac{ 1 }{ 1-x }\] and \[g(x) = \frac{ x-1 }{ x }\] are inverse functions by computing their compositions.

Answer 2

\[\frac{ 1 }{ 1-\frac{ (x-1) }{ x } }\]

Answer 3

\[\frac{ 1 }{ \frac{ x-x+1 }{ x } }\]

Answer 4

\[\frac{ x }{ 1 }\]

Answer 5

x

Answer 6

you've shown that

f(g(x)) = x

for all x in the domain

Answer 7

now you need to show if g(f(x)) = x is true

Answer 8

The answer on my answer sheet says "yes" but I don't know why something is yes or no.

Answer 9

oh ok.

Answer 10

The original question is "are f and g inverses of each other"

the answer is "Yes"

I'm assuming your teacher wants you to show why "yes" is the correct answer

Answer 11

\[\frac{ \frac{ 1 }{ 1-x }-1 }{ \frac{ 1 }{ 1-x } }\]

Answer 12

I don't know what would make the answer yes, though.

Answer 13

simplify that last expression you wrote

Answer 14

\[\frac{ \frac{ 1-(1-x) }{ 1-x } }{ \frac{ 1 }{ 1-x } }\]

Answer 15

\[\frac{ x(1-x) }{ 1-x }\]

Answer 16

x

Answer 17

So I got x both ways. Is that significant?

Answer 18

so because

f(g(x)) = x
g(f(x)) = x

this means that f and g are inverses of each other

Answer 19

So if I don't get x than they are not inverse?

Answer 20

if you got f(g(x)) to be equal to something other than x, then f and g wouldn't be inverses of each other.

or

if you got g(f(x)) to be equal to something other than x, then f and g wouldn't be inverses of each other.

Answer 21

So if I don't get x than they are not inverse?

Answer 22

if f and g are inverses of each other, then

f(g(x)) = x
and
g(f(x)) = x

for all defined x values in the domain

Answer 23

`So if I don't get x than they are not inverse?`
yes

Answer 24

Thank You!

Answer 25

no problem