Start with writing the function in standard form. Then find its vertex to graph it.
Inverse: f^(- 1)(x)=± sqrt(x)
Graph:
Practice makes perfect
Before we can find the inverse of the given function, we need to replace f(x) with y.
f(x)=x^2 ⇔ y=x^2
To algebraically determine the inverse of the given equation, we exchange x and y and solve for y.
Given Equation & Inverse Equation
y= x^2 & x= y^2
The result of isolating y in the new equation will be the inverse of the given function.
Now we have the inverse of the given function.
y=± sqrt(x)
Graphing the Function
Because the given function is a parabola, to graph it we should first determine its vertex. Notice that the function is in standard form, so let's start with highlighting the coefficients.
&Standard Form &&Function
&y= ax^2+ bx+ c &&y= 1x^2+ 0x+ 0
In this form, if a is positive, the parabola opens upward. If a is negative, the parabola opens downward. Since 1>0, the parabola of this function opens upward. To find the vertex, we first need to find the x coordinate of the vertex.
x=-b/2 a
Let's find it!
Thus, the vertex of the parabola is (0,0). To graph the parabola let's choose two more points, one on either side of the vertex. Let's use x=-2 and x=2. By substituting these coordinates into the function, we can find the y coordinates.
x
x^2
y
Point
-2
( -2)^2
4
(-2,4)
0
( 0)^2
0
(0,0)
2
( 2)^2
4
(2,4)
Let's plot the points and connect them to graph the parabola.
Graphing the Inverse of the Function
Finally, we can graph the inverse of the function by reflecting the parabola across y=x. This means that we should interchange the x- and y-coordinates of the points that are on the parabola.
