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Chapter 5: Q 52. (page 269)

The function fx=1-3xis one-to-one. Find its inverse and check the answer. Graph f,f-1 and y=x on the same coordinate axes.

Short Answer

Expert verified

Inverse of the function fx=1-3xis f-1x=1-x3. Graph is shown below.

Step by step solution

01

Step 1. Given information.

Given a one-to-one functionfx=1-3x.

02

Step 2. Find the inverse of the function f and verify.

To find the explicit form of the inverse solve x=1-3y for y.

It can be seen that y=1-x3.

Note that if fgx=x and gfx=x, then fand g are inverse of one another.

Prove ff-1x=xas follows.

ff-1x=f1-x3=1-31-x3=1-1+x=x

Prove f-1fx=xas follows.

f-1fx=f-11-3x=1-1-3x3=3x3=x

Therefore,f-1x=1-x3is inverse offx=1-3x.

03

Step 3. Graph f,f-1 and y=x on the same coordinate axes.

Graph fx=1-3x,f-1x=1-x3and y=xas follows.

Note the symmetry of the graphs with respect to the line y=x.

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