Question

@jim_thompson5910

Answer 1

Step 1) Start with f(x)

Step 2) Replace every x with g(x)

Step 3) Replace the g(x) on the right side with (x+7)/2


\[\Large f(x) = 2x - 7\]

\[\Large f(g(x)) = 2*g(x) - 7\]

\[\Large f(g(x)) = 2*\frac{x+7}{2} - 7\]

What is the next step?

Answer 2

I don't know :(

Answer 3

hint: something will cancel

Answer 4

the left side

Answer 5

Do you see how the '2's cancel here?

\[\Large f(g(x)) = 2*\frac{x+7}{2} - 7\]
\[\Large f(g(x)) = \cancel{2}*\frac{x+7}{\cancel{2}} - 7\]

Answer 6

yes

Answer 7

After the cancellation, we're left with
\[\Large f(g(x)) = x+7 - 7\]
which simplifies to what?

Answer 8

7x

Answer 9

7-7 turns into what?

Answer 10

0

Answer 11

so x+7-7 becomes x+0 which is just x

Answer 12

So all this means we have
\[\Large f(g(x)) = x\]

Answer 13

agreed so far?

Answer 14

yes

Answer 15

so we did half the work so far

Answer 16

we found f(g(x))

we now need to find g(f(x))

Answer 17

Step 1) start with g(x)

Step 2) Replace every x with f(x)

Step 3) Replace the f(x) on the right side with 2x-7


\[\Large g(x) = \frac{x+7}{2}\]
\[\Large g(f(x)) = \frac{f(x)+7}{2}\]
\[\Large g(f(x)) = \frac{2x-7+7}{2}\]
I'll let you simplify

Answer 18

if you're stuck, try to simplify `2x-7+7` first

Answer 19

hang on please

Answer 20

alright

Answer 21

the 7's cancel out

Answer 22

we're left with g(f(x))=2x-2

Answer 23

yes they combine to 0 and go away

Answer 24

\[\Large g(f(x)) = \frac{2x-7+7}{2}\]
\[\Large g(f(x)) = \frac{2x+0}{2}\]
\[\Large g(f(x)) = \frac{2x}{2}\]
what next?

Answer 25

g(f(x))=x

Answer 26

Because \(\Large f(g(x)) = x\) and \(\Large g(f(x)) = x\) (for all x in the domain), this means the functions f(x) and g(x) are inverses of each other.