Before we can find the inverse of the given function, we need to replace g(x) with y.
g(x)=(x+1)^2-3
⇕
y=(x+1)^2-3
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=( x+1)^2-3 & x=( y+1)^2-3
The result of isolating y in the new equation will be the inverse of the given function. Remember that after exchanging x and y our domain is y≥ -1.
Now we have the inverse of the given function.
g^(-1)(x)=sqrt(x+3)-1
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 graphing form, so let's highlight the coefficients.
&Vertex Form
&y= a(x- h)^2+ k [3mm]
&Function
& y = (x+1)^2-3
& ⇕
&y= 1(x-( -1))^2+( -3)
Thus, the vertex of the parabola is (-1,-3). To graph the parabola with the domain x≥-1, let's choose two more points on the right side of the vertex. Let's use x=1 and x=2. By substituting these coordinates into the function, we can find the y-coordinates.
x
y=(x+1)^2-3
y
Point
x= -1
y=( -1+1)^2-3
y=-3
(-1,-3)
x= 1
y=( 1+1)^2-3
y=1
(1,1)
x= 2
y=( 2+1)^2-3
y=6
(2,6)
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.
Points
Reflection across y=x
( -1, -3)
( -3, -1)
( 1, 1)
( 1, 1)
( 2, 6)
( 6, 2)
Stating the Domain and Range of the Inverse Function
To determine the domain of the inverse function, the radicand cannot be negative. This means that the radicand must be greater than or equal to 0. Let's find the value(s) of x that make this true.
All values of x that are greater than or equal to -3 are included in the domain.
Domain
x≥ -3
For the range, think about the fact that the principal root of a number is always non-negative. To obtain the formula of the inverse function, we need to subtract 1 from both sides of the inequality.
sqrt(x+3) ≥ 0
⇕
sqrt(x+3) - 1 ≥ -1
Therefore, the range is all values of y such that y≥ -1.
Range
y≥ -1
