Question
(c) The original function \( y=A(r) \) converted an input, the circle's radius, to an output, the circle's area. What are the inputs and outputs of the inverse function? Input: REASOLput: 8. The domain and range of a one-to-one function, \( y=f(x) \), are given below in set-builder notation. Giv the domain and range of this function's inverse also in set-builder notation. \[ y=f(x) \] Domain: \( \{x \mid-3 \leq x<5\} \) Range: \( \{y \mid y>-2\} \)
Ask by Graham Fuentes. in the United States
Feb 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
Part (c):
- **Input of the inverse function**: Circle's area
- **Output of the inverse function**: Circle's radius
Part 8:
- **Domain of the inverse function**: \( \{ x \mid x > -2 \} \)
- **Range of the inverse function**: \( \{ y \mid -3 \leq y < 5 \} \)
Solution
Part (c):
The original function A takes a circle’s radius as its input and returns its area as the output. The inverse function reverses this process; it takes a circle’s area as its input and returns the corresponding circle’s radius as its output.
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Part 8:
The function f has
Domain: { x ∣ –3 ≤ x < 5 } and Range: { y ∣ y > –2 }.
For the inverse function, the domain and range are swapped. That is,
Domain of f⁻¹: { x ∣ x > –2 } and Range of f⁻¹: { y ∣ –3 ≤ y < 5 }.
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Simplify this solution Mind Expander
When you flip the function to find its inverse, the inputs become the outputs of the original function and vice versa. So, for the inverse of your area function \( y = A(r) \), the input will be the area \( A \) and the output will be the radius \( r \). Now, if we take the domain and range you've given, the domain of the inverse function will be the range of the original, which is \( \{y \mid y > -2\} \), and the range of the inverse function will be the domain of the original, \( \{x \mid -3 \leq x < 5\} \). Fun fact: flipping the inputs and outputs like this is one of the coolest tricks in math!
