Question

Describe how this graph changed from the graph or the recriprocal function g(x)=1/x?

f(x)=2x-1 / x+3

I have the answers, I just don't know how to get all of them.

Answer 1

\(\color{blue}{\displaystyle f(x)=\frac{2x-1}{x+3}=\frac{2x+6-7}{x+3}=\frac{2x+6}{x+3}-\frac{7}{x+3}=-\frac{7}{x+3}+2}\)

\(\color{red}{\displaystyle t=x+3}\)

\(\color{blue}{\displaystyle f(t)=-\frac{7}{t}+2}\)

\(\color{blue}{\displaystyle \frac{-f(t)}{7} =\frac{1}{t}-\frac{2}{7}}\)

\(\color{blue}{\displaystyle \frac{-f(t)}{7}+\frac{2}{7} =\frac{1}{t}-\frac{2}{7}+\frac{2}{7}}\)

\(\color{blue}{\displaystyle \frac{-f(t)}{7}+\frac{2}{7} =\frac{1}{t}}\)

Answer 2

So, if you need the order of operations to get f(x) from g(x), then you can follow the steps backwards. (I will start for you.)

So, first I subtracted -2/7. (Shift down by 2/7 units)
Then, I scaled the function by a factor of 7, and reflected across the x-axis.
Then, shifted the function 3 units left. (Finally obtaining f(x).)