Can you please explain in full detail. 4. Consider the function f(x) = |x| – 1.(a) Graph f(x) below. What is the domain of f(x), and how can we need restrict the domain of f to make it invertible?Why do we need to restrict the domain to those values?(b) Graph the inverse of f(x) on your restricted domain from part (b). Explain how you determined where to placepoints.

Answered: Consider the function f(x) = |x| – 1.…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section5.1: Inverse Functions

Problem 56E

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Can you please explain in full detail.

4. Consider the function f(x) = |x| – 1.
(a) Graph f(x) below. What is the domain of f(x), and how can we need restrict the domain of f to make it invertible?
Why do we need to restrict the domain to those values?
(b) Graph the inverse of f(x) on your restricted domain from part (b). Explain how you determined where to place
points.

Expert Solution

Step 1

(a) Graph the function f(x) below.

$T h e f u n c t i o n f (x) i s d e f i n e d f o r a l l v a l u e s f o r x . T h e r e f o r e, d o m a i n f o r f (x) i s x \in R . T h e f u n c t i o n f (x) i s s t r i c t l y i n c r e a \sin g f o r x > 0 a n d s t r i c t l y d e c r e a \sin g f o r x < 0 . S o, d o m a i n f o r f u n c t i o n f (x) m u s t b e s t r i c t l y i n c r e a \sin g o r d e c r e a \sin g f o r i n v e r s e f u n c t i o n t o e x i s t . T h u s, c o n s i d e r t h e i n c r e a \sin g p a r t o f t h e f u n c t i o n f (x) f o r d e t e r m i n i n g t h e i n v e r s e . H e n c e, d o m a i n f o r f t o m a k e i t i n v e r t i b l e i s x \in [0, \infty) . T h e d o m a i n f o r f i s r e s t i r c t e d b e c a u s e i t i s n o t p o s s i b l e t o d e t e r m i n e t h e i n v e r s e f o r a m a n y - o n e f u n c t i o n .$

Step 2

(b) Now, determine the inverse for f(x) for x>0.

$\begin{array}{rcl} f (x) & = & |x| - 1 \\ F o r x & > & 0, f (x) = x - 1 \\ y & = & x - 1 \\ r e p l a c e x & \leftrightarrow & y \\ S o, x & = & y + 1 \\ A g a i n r e p l a c e, x & \leftrightarrow & y \\ y & = & x + 1 \\ S o, f^{- 1} & = & x + 1, f o r x \in [- 1, \infty) \end{array}$

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