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icang [17]
1 year ago
9

I NEED HELP WITH QUESTION PLEASE!! What is the inverse of f(x) (x) = (x-4)² for x ≥ 4?

Mathematics
1 answer:
goblinko [34]1 year ago
5 0

Answer:

4th choice \bold{f^{-1}(x) = \sqrt{x} + 4}

Step-by-step explanation:

<u>Definition of the inverse of a function</u>

A function g is the inverse of a function f if whenever y=f(x) then x=g(y). In other words, applying f and then g is the same thing as doing nothing. We can write this in terms of the composition of f and g as g(f(x))=x. The domain of f becomes the range of g and the range of f becomes the domain of g

To solve for the inverse of the function f(x) =\left(x-4\right)^2

Let y=\left(x-4\right)^2

\mathrm{Replace}\:x\:\mathrm{with}\:y \text{ and replace }\:y\:\mathrm{with}\:x

x=\left(y-4\right)^2

Switch sides
\left(y-4\right)^2=x

Take square roots on both sides
y-4=\pm\sqrt{x}

Add 4 on both sides to solve for y

y = \pm\sqrt{x} + 4

We have two solutions

y=\sqrt{x}+4,\:y=-\sqrt{x}+4

To determine which one of these to be chosen not that in the given choices we can eliminate the first two since x cannot be negative

The third choice can also be eliminated since
-\sqrt{x} + 4 is a decreasing function for x \ge 0

So the last answer choice is correct and the inverse off(x) = (x-4)^2

is given by  f^{-1}(x) = \sqrt{x} + 4

Answer:4th choice \bold{f^{-1}(x) = \sqrt{x} + 4}

Note

Domain of  (x-4)² is [4, ∞) since x ≥ 4 and (x-4)²  cannot be negative

Range of (x-4)²  is [0,  ∞)

Domain of \sqrt{x}\:+\:4 is [0, ∞)

Range of \sqrt{x}\:+\:4 is [4, ∞)

so indeed the domain of (x-4)² has become the range of \sqrt{x}\:+\:4 and the range of (x-4)² has become the domain of \sqrt{x}\:+\:4

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