Question

Please help solve the question below. It's a little complicated so I'll have to use the equation generator here to be able to show it to you. When you get your answer, could you please show me? Thank you!

Answer 1

Here's the question.
For \[f(x)=\sqrt{x-3},\] find \[(f o f^-1)(4)\]

Answer 2

A. 4
B. 67
C. -4
D. 343

Answer 3

The composition of a function and it's inverse just returns the initial input. Think about multiplication and division as inverses. If I multiply a number by 10 and then divide the result by 10 I will get my original number. In your case if you generate the composite function like I showed in the last problem your will get the composite is = to x. If you need help generating the inverse of a function let me know.

Answer 4

@hullsnipe I understand inverse, well I should, but I think I will need help with the inverse of a function.

Answer 5

To find the inverse of a function there are 4 steps
\[f(x)=\sqrt{x-3}\]
STEP 1: Replace f(x) with y
\[y=\sqrt{x-3}\]
STEP 2: Swap x and y
\[x=\sqrt{y-3}\]
STEP 3: Solve for y
\[x ^{2}=y-3\]
\[y=x ^{2}+3\]
STEP 4: Replace y with \[f ^{-1}(x)\]
\[f ^{-1}(x)=x ^{2}+3\]

Answer 6

Now your composition of the function and inverse will look like the following
\[f(f ^{-1}(x))=\sqrt{(x ^{2}+3)-3}\]
When you simplify the right hand side your get:
\[f(f ^{-1}(x))=x\]

Answer 7

@hullsnape
On the right hand side, I got \[\sqrt{16}\] so I guess the answer is 4.
Is this correct?

Answer 8

@hullsnipe

Answer 9

yes. The composition of a function and it's inverse return the original value. If you recognize this then now work required INPUT=4 therefore OUTPUT =4.

Answer 10

Thank you so much again! I don't know if I was asleep during class when we talked about this or what! :) You really helped me out.

Answer 11

@hullsnipe