SOLUTION: Find the inverse function of f:[3,∞) -> R, f(x) = (x-4)^2 +1 Why does the domain of f need to be restricted to [4,∞) in this question?

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Question 1028867: Find the inverse function of f:[3,∞) -> R, f(x) = (x-4)^2 +1
Why does the domain of f need to be restricted to [4,∞) in this question?
Found 2 solutions by jim_thompson5910, josgarithmetic:
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!
Here is the basic outline for finding the inverse
A) Replace f(x) with y (step 2 in table below)
B) Swap x and y (step 3 in table below)
C) Solve for y (steps 4 through 7 in table below)
Let's use this to find the inverse of the given function
NumberStatementReason/Explanation
1.None needed. This is the original function given
2.Replace f(x) with y
3.Swap x and y. Now we isolate y.
4.Subtract 1 from both sides
5. Flip the equation
6.Take the square root of both sides. See note below
7.Add 4 to both sides


Note: the domain of [4,infinity) for f(x) turns out to be the range of the inverse function. Domain of original = range of inverse. In order to stretch onto positive infinity, we need to use the plus version of the plus/minus. So instead of using plus/minus, we can just use plus all by itself. So instead of using we stick with

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The inverse function is therefore

The domain of f(x) must be restricted to [4,infinity) because including 3 in the domain makes the function not one-to-one. Notice how f(3) = f(5) = 2. You must restrict the domain to make f(x) one-to-one in order for f(x) to be invertible to a function.
Answer by josgarithmetic(39618)   (Show Source): You can put this solution on YOUR website!
Look first for the inverse relation to f.

for the inverse function of f.




The domain of R(x) is for TWO different functions. R itself is a relation and not a function. This is because the "plus or minus" part of the expression. The domain for either branch of R is .

Look again at function f(x). Domain is ALL REAL NUMBERS. What about the range of f(x)? f(x) is a parabola with a vertex minimum value at (4,1). This means that the RANGE for f(x) is , or as a description, from 4 inclusive toward infinity.

Going from a function (f(x)) to its inverse, the domain and range switch. The range for f(x) is the DOMAIN for either branch of R(x).



Function f(x)


Upper branch of R(x)


Lower branch of R(x)
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