Consider the function f(x)=2x + 5. Find the inverse of f(x) and name it g(x). Show and explain Use function composition to show that f(x) and g(x) are inverses of each other. Show and explain your...
Consider the function f(x)=2x + 5.
Find the inverse of f(x) and name it g(x). Show and explain
Use function composition to show that f(x) and g(x) are inverses of each other. Show and explain your work. (Hint: Find (f ◦ g)(x) and (g ◦ f)(x).)
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(a) `f(x)=2x + 5`
`y = 2x + 5`
To determine the inverse function, interchange x and y.
`x=2y+5`
Then, isolate y.
`2y=x-5`
`y=(x-5)/2`
Replace y with `f^(-1)(x)` , to indicate that it is the inverse of the given function.
`f^(-1)x = (x-5)/2`
(b) To check that `f^(-1)x = (x-5) / 2` is the inverse of `f(x)= 2x+5` , let
the inverse function be `g(x) = (x-5)/2` .
Note that if `(f o g (x) )= (g o f)(x)` , then f(x) and g(x) are inverses of each other.
So,
` (f o g (x) ) = (g o f)(x)`
`f ( g(x) ) = g(f(x))`
At the left side, replace the x in f(x) with (x-5)/2. And at the right side, replace the x in g(x) with 2x+5.
`2((x-5)/2) +5 = (2x+5-5)/2`
`x-5 + 5 = 2x/2`
`x = x`
Hence, `g(x) = (x-5)/2` is the inverse function of `f(x) = 2x +5` .
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