How To Find The Inverse Of A Quadratic Function
INVERSE OF A QUADRATIC FUNCTION
The full general grade of a quadratic function is
f(ten) = ax2 + bx + c
Then, the inverse of the above quadratic part is
f-1 (10)
For example, let us consider the quadratic function
g(x) = x2
So, the changed of the quadratic function is g(x) = x 2 is
g(x) -1 = √x
Finding inverse of a quadratic function :
Allow f(10) be a quadratic function.
Step 1 :
Replace f(x) past y and interchange the variables x and y.
Step two :
Solve for y and supercede y by f-ane(x).
Case i :
Find the inverse of the quadratic function and graph it.
f(x) = x2
Solution :
Replace f(x) by y.
y = 10 2
Interchange x and y.
10 = y 2
y 2 = ten
Solve for y.
Take square root on both sides.
y = ±√x
Supercede y by f-1(x).
f-1(x) = ±√x
Graphing the inverse of f(10) :
We can graph the original part past plotting the vertex (0, 0). The parabola opens up, because a is positive.
And we get f(1) = ane and f(2) = 4, which are likewise the same values of f(-ane) and f(-2) respectively.
To graph f-1(x) , we have to take the coordinates of each point on the original graph and switch the x and y coordinates.
For instance, (2, four) becomes (4, 2).
Nosotros take to do this because the input value becomes the output value in the inverse, and vice versa.
The graph of the changed is a reflection of the original part almost the line y = x.
Example 2 :
Find the inverse of the quadratic part and graph information technology.
f(10) = 2(x + 3)2 - four
Solution :
Replace f(x) by y.
y = 2(x + 3)2 - 4
Interchange ten and y.
10 = two(y + 3) 2 - 4
Solve for y.
ten + 4 = 2(y + three)two
(ten + 4)/2 = (y + three) 2
Take square root on both sides.
±√[(10 + four)/2] = y + 3
±√[(x + 4)/2] - 3 = y
y = -3 ± √[(10 + 4)/2]
Replace y past f -1 (x).
f -1 (x) =-3 ± √[(x + 4)/ii]
Graphing the inverse of f(ten) :
We tin can graph the original part by plotting the vertex (-3, -iv). The parabola opens upward, because a is positive.
And we get f(-ii) = -2 and f(-1) = 4, which are also the same values of f(-iv) and f(-five) respectively.
To graph f-i(10) , we have to take the coordinates of each betoken on the original graph and switch the ten and y coordinates.
For example, (-one, iv) becomes (4, -1).
We accept to do this because the input value becomes the output value in the changed, and vice versa.
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