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Math  /  Algebra

QuestionThe domain of a one-to-one function ff is [1,∞)[1, \infty), and its range is [−3,∞)[-3, \infty). State the domain and the range of f−1f^{-1}.
What is the domain of f−1\mathrm{f}^{-1} ? The domain of r−1\mathrm{r}^{-1} is [−3,∞)[-3, \infty). (Type your answer in interval notation.) What is the range of f−1f^{-1} ? The range of f−1f^{-1} is □\square (Type your answer in interval notation.)

Studdy Solution

STEP 1

1. The function f f is one-to-one.
2. The domain of f f is [1,∞)[1, \infty).
3. The range of f f is [−3,∞)[-3, \infty).
4. We need to find the domain and range of the inverse function f−1 f^{-1} .

STEP 2

1. Understand the relationship between a function and its inverse.
2. Determine the domain of f−1 f^{-1} .
3. Determine the range of f−1 f^{-1} .

STEP 3

For a one-to-one function f f , the domain of its inverse f−1 f^{-1} is the range of f f , and the range of f−1 f^{-1} is the domain of f f .

STEP 4

Since the range of f f is [−3,∞)[-3, \infty), the domain of f−1 f^{-1} is:
Domain of f−1=[−3,∞) \text{Domain of } f^{-1} = [-3, \infty)

STEP 5

Since the domain of f f is [1,∞)[1, \infty), the range of f−1 f^{-1} is:
Range of f−1=[1,∞) \text{Range of } f^{-1} = [1, \infty)
The domain of f−1 f^{-1} is [−3,∞)[-3, \infty) and the range of f−1 f^{-1} is [1,∞)[1, \infty).

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