Given the function f(x) = (x- 14)², xs 14, (a) Find f (x). (b) Graph f and f1 in the same rectangular coordinate system. (c) Use interval notation to give the domain and the range of f and f1.
Given the function f(x) = (x- 14)², xs 14, (a) Find f (x). (b) Graph f and f1 in the same rectangular coordinate system. (c) Use interval notation to give the domain and the range of f and f1.
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![Given the function \( f(x) = (x - 14)^2 \), \( x \leq 14 \):
**(a)** Find \( f^{-1}(x) \).
**(b)** Graph \( f \) and \( f^{-1} \) in the same rectangular coordinate system.
**(c)** Use interval notation to give the domain and the range of \( f \) and \( f^{-1} \).
**(a)** Find \( f^{-1}(x) \). Be sure to include any domain restrictions.
\[ f^{-1}(x) = \sqrt{x} + 14,\ x \geq 0 \]
**(b)** Choose the correct graph which shows \( f \) and \( f^{-1} \) graphed in the same coordinate system.
- Option A shows a graph with a parabola and a linear function.
- Option B shows a similar graph to A.
- Option C also displays a parabola and a linear line.
- Option D contains graph elements similar to others.
The correct graph must accurately reflect the parabolic shape of \( f(x) \) and the linear form of its inverse, with correct intersection points and symmetry about the line \( y = x \).
**Graph Analysis**:
- The parabola \( f(x) \) is represented as a U-shaped curve opening upwards.
- The linear function \( f^{-1}(x) \) should be reflected across the line \( y = x \).
- Each graph option shows different variations of these plots; select the one that correctly reflects the inverse relationship with the correct domain and range constraints.
The solution involves analyzing the symmetry and intersections between \( f(x) \) and \( f^{-1}(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0c96805e-145e-4f92-a1ca-eaa2329609ab%2F7636e5ff-2a0a-481a-869a-bb56f3f1096f%2Fr8y70y7_processed.png&w=3840&q=75)
Transcribed Image Text:Given the function \( f(x) = (x - 14)^2 \), \( x \leq 14 \):
**(a)** Find \( f^{-1}(x) \).
**(b)** Graph \( f \) and \( f^{-1} \) in the same rectangular coordinate system.
**(c)** Use interval notation to give the domain and the range of \( f \) and \( f^{-1} \).
**(a)** Find \( f^{-1}(x) \). Be sure to include any domain restrictions.
\[ f^{-1}(x) = \sqrt{x} + 14,\ x \geq 0 \]
**(b)** Choose the correct graph which shows \( f \) and \( f^{-1} \) graphed in the same coordinate system.
- Option A shows a graph with a parabola and a linear function.
- Option B shows a similar graph to A.
- Option C also displays a parabola and a linear line.
- Option D contains graph elements similar to others.
The correct graph must accurately reflect the parabolic shape of \( f(x) \) and the linear form of its inverse, with correct intersection points and symmetry about the line \( y = x \).
**Graph Analysis**:
- The parabola \( f(x) \) is represented as a U-shaped curve opening upwards.
- The linear function \( f^{-1}(x) \) should be reflected across the line \( y = x \).
- Each graph option shows different variations of these plots; select the one that correctly reflects the inverse relationship with the correct domain and range constraints.
The solution involves analyzing the symmetry and intersections between \( f(x) \) and \( f^{-1}(x) \).
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