Question

PLEASE HELP!!! 45 PTS AND BRAINLIEST!!!

1) Complete the work for each method below:

Method A: Given f(x) = 3x - 4 and g(x) = x+4/3
Show that these are inverse functions by finding f^-1 (x) and showing that it is the same as g(x)

Method B: Given f(x) = 3x - 4 and g(x) = x+4/3
Show that these are inverse functions by showing that when the output of one function is used for the input of the other function, the final output is equal to the original input value. (you may choose any initial input)

Method C: Given f(x) = 3x - 4 and g(x) = x+4/3
Verify that these are the inverse function by showing that f(g(x)) = x AND g(f(x)) = x

Answer 1

Answer:

See explanation!

Step-by-step explanation:

Let us first give some principle theory to aid our solution.

Considering two functions $A(x)$ and $B(x)$, in order to show that function

which is the inverse of $y=ax$.

Now let as solve our problem. We are given the following:

$f(x)=3x-4\\g(x)=\frac{x+4}{3}$

Method A: Show that these are inverse functions by finding f^-1 (x) and showing that it is the same as g(x).

Let us take $[f(x)=y]=3x-4$ and "exchanging" our variables we have

$x=3y-4\\x+4=3y\\\\y=\frac{x+4}{3}$

which is exactly the same with our given function of $g(x)=\frac{x+4}{3}$, so proved!

Method B: Show that these are inverse functions by showing that when the output of one function is used for the input of the other function, the final output is equal to the original input value. (you may choose any initial input)

For this case we will use a simple input let us say $x=1$. Thus taking the $f(x)$ function and plugging in we have:

$f(x=1) = 3(1)-4\\f(1)=3-4\\f(1)=-1$

Now let us take the output of $f(1)$ which is $-1$ and use it the input to our second function of $g(x)$, so we have:

$g(x=-1) = \frac{(-1)+4}{3}\\ \\g(-1)=\frac{3}{3}\\ \\g(-1)=1$

so the output of the second function is equal to the original input value of the first function, hence proved!

Method C: Verify that these are the inverse function by showing that f(g(x)) = x AND g(f(x)) = x.

Basically we are asked to prove that both $f(g(x))=g(f(x))=x$

To do so, we just replace one function into the $x$ value of the other function as follow:

$f(g(x))=3(\frac{x+4}{3} )-4\\\\f(g(x))=x+4-4\\\\f(g(x))=x$

Lets repeat now for the opposite as follow:

$g(f(x))=\frac{(3x-4)+4}{3}\\ \\g(f(x))=\frac{3x}{3}\\ \\g(f(x))=x$

Hence proved!