Pt1420 Unit 1

Hallo, Professor Hanson, and readers of this Journal Entry
Problem 1
Let \( \ g: \mathrm{R} \rightarrow \mathrm{R} \) be defined by \( \ g(x) = x^{2} \)
Task 1a
What is \( \ g^{-1}(4) ? \)
The Process
Firstly, let us find \( \ g^{-1}(x) \). As we know the inverse will be undoing what \( \ g \) has done to \( \ x \) using the following steps
↓
Step 1. We write down the function
\( \ g(x) = x^{2} \Leftrightarrow y = x^{2} \)
\( \ y = x^{2} \)
↓
Step 2. We interchange variables by replacing the occurrence of y with x, and x with y
\( \ y = x^{2} \rightarrow x = y^{2} \)
\( \ x = y^{2} \)
↓
Step 3. Now we solve for y
\( \ x = y^{2} \rightarrow \sqrt{x} = y \)
\( y = \sqrt{x} \Leftrightarrow g^{-1}(x) = \sqrt{x} \)
↓
Then now, the inverse of \( \ g(x) \) is

Given that \( \ r : \mathrm{R} \rightarrow \mathrm{Z} \) Where \( \ r(x) = |x| \)
What is \( \ r^{-1}(1) ? \)
The Process
This problem will be solved in a similar fashion as in Task 1a, only with a slight difference because of the absoluteness of the image of \( \ 2 \) under function \( \ r \).
In this case the functiion
\( \ r(x) = |x| \)
Has done both

\( \ r(x) = -x \) And \( \ r(x) = x \)

Using the inverse steps introduced in Task 1a the process will be as follows

\( \ r(x) = -x \)
We re-write this as
\( \ r(x) = -x \Rightarrow \ y = -x \)
\( \ y = -x \)
Then interchange variable
\( \ x = -y \)
Therefore, solve for y
\( \ y = -x \Leftrigharrow \ r^{-1}(x) = -x \)
\( \ r^{-1}(x) = -x