What can be said about fof^-1(x) of the function: y = x/4 + 3.
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The notation of a circle between two functions (for example, f o g (x)) means the composition of two functions. It is found as function f (the one that comes first) of g(x):
f o g (x)= f(g(x)).
In this case, the two functions in question are the function f and its inverse, denoted as f^(-1). (Note that the exponent of -1 here means inverse, NOT the reciprocal, which would be 1/f.)
Since the inverse of a function is created by swapping the independent and dependent variables, it takes the value of f as the independent variable and converts it to original input variable x, making it dependent variable. If the function and its inverse are composed, the result is the original variable, x:
f o f ^(-1) (x)= f(f^(-1)(x)) = x.
In other words, the composition of a function and its inverse (or vice versa) is the identity function: it equals x.
For the given function y = x/4 + 3, this can be confirmed by obtaining the inverse function explicitly and then finding the composition.
The inverse function is obtained by exchanging x and y, and then solving for y:
x = y/4 + 3
x - 3 = y/4
y = 4(x - 3)
Now, if the original function f(x) = x/4 + 3 is evaluated at 4(x-3) instead of x, we get
f o f^(-1) (x) = [4(x - 3)]/4 + 3 = x - 3 + 3 = x, as expected.
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fof^-1(x)
= f(f^-1(x))
= x, by definition
Here y = x/4 + 3
The inverse function is found by expressing y in terms of x, x = (y - 3)*4. Now interchange y and x. f^-(x) = (x - 3)*4
fof^-1(x)
= f((x - 3)*4)
= (x - 3)*4/4 + 3
= x
This proves that for f(x) = x/4 + 3, fof^-1(x) = x
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
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