Find the inverse of y = x/4 + 3. What can be said about fof^-1(x) of any function.

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We are given the function y = x/4 + 3 and have to find the inverse.

=> y = x/4 + 12/4

=> y = (x + 12)/4

=> 4y = x +12

=> x = 4y – 12

interchange x and y

=>** y = 4x – 12**

Now let the function f(x) = x/4 + 3 and g(x) = 4x – 12

We see that f (g(x)) = f (4x – 12) = (4x -12)/4 + 3 = x

Also g (f(x)) = g(x/4 + 3) = 4(x/4 +3) – 12 = x

**We see that for any function f(x), the function of the inverse function of x or fof^-1(x) is always equal to x.**

To determine the inverse, we'll have to write a expression of a function, with respect to y, starting from original function.

We'll write the given function:

y = x/4 + 3

We'll multiply by 4 both sides:

4y = x + 12

We'll use the symmetric property:

x + 12 = 4y

We'll isolate x to the left side. For this reason, we'll subtract 12 both sides:

x = 4y - 12

**The inverse function is:**

**f^-1(x) = 4x - 12**

Now, we'll compose the functions:

(fof^-1)(x) = f(f^-1(x))

We'll substitute x by the f^-1(x) in the expression of f(x):

f(f^-1(x)) = f^-1(x)/4 + 3

We'll substitute f^-1(x) by it's expression:

f(f^-1(x)) = (4x - 12)/4 + 3

f(f^-1(x)) = 4x/4 - 12/4 + 3

f(f^-1(x)) = x - 3 + 3

We'll eliminate like terms and we'll get:

**f(f^-1(x)) = x**

To find the inverse of y = x/4 + 3. What can be said about fof^-1(x) of any function.

Let f(x) = x/4+3.

Let f^-1(x) = y.

Then by definition x = f(y)

Therefore x = y/4 +3....(1), as f(x) = x/4+3.

So we solve for y from eq (1):

x= y/4 +3. We multiply both sides by 4.

4x= y+12. We subtract 1 from both sides.

4x-12 = y.

Or y = 4x-12.

Therefore y = 4x-12 is the inverse of y = x/4+3.

To find fof^-1:

y = f(x) = x/4+3. y = f^-1(x) = 4x-12.

fof^-1(x) = f ( f^-1(x)) = 4{f^-1(x)}+3

fof^-1(x) = f((f^-1(x)) = (4x-12)/3+3.

fof^-1(x) = f((f^-1(x)).

fof^-1(x) = f((f^-1(x)) = x-3+3.

fof^-1(x) = f((f^-1(x)) = x.

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