Answer
(a) $$f^{-1}(x)=x-1$$
The graphs are in the image below.
(b) $$f^{-1}(x)=x-b$$
The graphs of $f$ and $f^{-1}$ are parallel to each other.
(c) We can conclude that the inverse of those functions whose graphs are parallel to the line $y=x$:
- Their graphs are also parallel to the line $y=x$ and to the graphs of the original functions.
- They lie across the line $y=x$ from the original graphs, and their distance to the line $y=x$ is equal to that of the original graphs.
Work Step by Step
(a) $$y=f(x)=x+1\hspace{1cm}$$
- To find its inverse:
1) Solve for $x$ in terms of $y$:
$$y=x+1$$ $$x=y-1$$
2) Interchange $x$ and $y$:
$$y=x-1$$
Therefore, $$f^{-1}(x)=x-1$$
The graphs are shown in the image below.
(b) $$y=f(x)=x+b\hspace{1cm}$$
- To find its inverse:
1) Solve for $x$ in terms of $y$:
$$y=x+b$$ $$x=y-b$$
2) Interchange $x$ and $y$:
$$y=x-b$$
Therefore, $$f^{-1}(x)=x-b$$
Looking at the formula of $f$ and $f^{-1}$, since both have slope $1$, their graphs would be parallel with each other.
(c) We see in (a) that the graph of $f^{-1}$ is across the other side of line $y=x$ from the graph of $f$.
And in fact, in (b), the same thing would happen. They are not just on opposite sides, but the distance from them to line $y=x$ are equal to the other.
So we can conclude that the inverse of those functions whose graphs are parallel to the line $y=x$:
- Their graphs are also parallel to the line $y=x$ and to the graphs of the original functions.
- They lie across the line $y=x$ from the original graphs, and their distance to the line $y=x$ is equal to that of the original graphs.
