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Math  /  Algebra

QuestionConsider the function f(x)=x−5+8f(x)=\sqrt{x-5}+8 for the domain [5,∞)[5, \infty). Find f−1(x)f^{-1}(x), where f−1f^{-1} is the inverse of ff. Also state the domain of f−1f^{-1} in interval notation. f−1(x)=f^{-1}(x)= □\square for the domain □\square

Studdy Solution

STEP 1

1. The function f(x)=x−5+8 f(x) = \sqrt{x-5} + 8 is defined for the domain [5,∞)[5, \infty).
2. We need to find the inverse function f−1(x) f^{-1}(x) .
3. We need to determine the domain of f−1(x) f^{-1}(x) in interval notation.

STEP 2

1. Set y=f(x) y = f(x) and solve for x x in terms of y y .
2. Express the inverse function f−1(x) f^{-1}(x) .
3. Determine the domain of f−1(x) f^{-1}(x) .

STEP 3

Set y=f(x) y = f(x) :
y=x−5+8 y = \sqrt{x-5} + 8

STEP 4

Isolate the square root term:
y−8=x−5 y - 8 = \sqrt{x-5}

STEP 5

Square both sides to eliminate the square root:
(y−8)2=x−5 (y - 8)^2 = x - 5

STEP 6

Solve for x x :
x=(y−8)2+5 x = (y - 8)^2 + 5

STEP 7

Express the inverse function:
Since y=f(x) y = f(x) , then x=f−1(y) x = f^{-1}(y) . Therefore, the inverse function is:
f−1(x)=(x−8)2+5 f^{-1}(x) = (x - 8)^2 + 5

STEP 8

Determine the domain of f−1(x) f^{-1}(x) :
Since the range of f(x) f(x) is [8,∞)[8, \infty) (as x−5\sqrt{x-5} is non-negative and starts from 0), the domain of f−1(x) f^{-1}(x) is the same as the range of f(x) f(x) .
Thus, the domain of f−1(x) f^{-1}(x) is [8,∞)[8, \infty).
The inverse function and its domain are:
f−1(x)=(x−8)2+5 f^{-1}(x) = (x - 8)^2 + 5
for the domain [8,∞)[8, \infty).

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