The graphs of a function and its inverse are symmetric to each other with respect to the line y=x.
Inverse function: f^(-1)(x)=(x-1)^2+3 for x≥ 1 Graph:
Practice makes perfect
Before we find the inverse of the given function, we should rewrite the function as an equation relating x and y.
f(x)=2sqrt((x-3)/4)+1
⇓
y=2sqrt((x-3)/4)+1
To algebraically determine the inverse of the given relation, we exchange x and y and solve for y.
c|c
Given Equation & Inverse Equation [0.8em]
y=2sqrt(( x-3)/4)+1 & x=2sqrt(( y-3)/4)+1
The result of isolating y in the new equation will be the inverse of the given function. Since this is a radical equation, let's first isolate the radical expression on one side of the equation.
Now, since x-1 equals a principal square root, it has to be greater than or equal to 0.
x-1≥0 ⇒ x≥ 1
Therefore, x has to be greater than or equal to 1. Let's continue solving.
Now that we have isolated y, we have found the inverse of the given function.
y=(x-1)^2+3 ⇒ f^(-1)(x) = (x-1)^2+3
Graphing the Function
Because the given function is radical, to graph it we should first determine its domain. To do so, recall that the radicand of a square root is always greater than or equal to 0.
(x-3)/4≥ 0 ⇔ x≥ 3
The domain of the given function is all real numbers greater than or equal to 3. With this in mind, we will make a table of values to graph the function.
x
2sqrt((x-3)/4)+1
y=2sqrt((x-3)/4)+1
3
2sqrt(( 3-3)/4)+1
1
4
2sqrt(( 4-3)/4)+1
2
7
2sqrt(( 7-3)/4)+1
3
12
2sqrt(( 12-3)/4)+1
4
Let's plot and connect the obtained points. Keep in mind that the domain is all real numbers greater than or equal to 3.
Graphing the Inverse of the Function
Finally, we can graph the inverse of the function by reflecting the graph of the given function across y=x. This means that we should interchange the x- and y-coordinates of the points that are on the graph.
