Explanation:
The original function is f(x):
[tex]f(x)=2(x-1)^3+1[/tex]
And g(x) is the transformed function after a reflection across the y-axis, a translation of 3 units right and 1 unit up.
Step 1. A reflection across the y-axis is represented by:
[tex]f(x)\rightarrow f(-x)[/tex]
Step 2. A translation of 3 units to the right is represented by:
[tex]f(x)\rightarrow f(x+3)[/tex]
Step 3. And a translation of 1 unit up is represented by:
[tex]f(x)\rightarrow f(x)+1[/tex]
Step 4. Combining these three transformations:
[tex]f(x)\rightarrow f(-x+3)+1[/tex]
Step 5. Function g(x) is defined as follows:
[tex]g(x)=f(-x+3)+1[/tex]
Which applied to the f(x) function is:
[tex]\begin{gathered} f(x)=2(x-1)^{3}+1 \\ \downarrow \\ g(x)=f(-x+3)+1 \\ \downarrow \\ Applying\text{ the transformation:} \\ g(x)=2(-x+3-1)^3+1+1 \end{gathered}[/tex]
Simplifying the operations:
[tex]g(x)=2(-x+2)^3+2[/tex]
That is the rule for function g(x).
Answer:
[tex]g(x)=2(-x+2)^{3}+2[/tex]
