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Q01 |
- What is inverse of a function ?
- Q02 | - Inverse of Y = a*x^2 + b*x + c is X = a*y^2 + b*y + c
- Q03 | - Find intersection of y = a*x^2 + b*x + c with it inverse
- Q04 | - Find intersection of y = x^2 - 2*x + 4 and its inverse
- Q05 | - Find intersection of y = x^2 - 3*x + 4 and its inverse
- Q06 | - Diagram : y = a*x^2 + b*x + c and its inverse
- Q02 | - Inverse of Y = a*x^2 + b*x + c is X = a*y^2 + b*y + c
Q01. Inverse function of y = a*x^2 + b*x + c
Definition
- 1. It is the reflection of Y = F(X) about Y = X
- 2. It is the image of Y = F(X) about Y = X (it looks like mirror)
- 3. The inverse is x = a*y^2 + b*y + c
- Y = F(X) and find X interms of Y
- That is to interchange X and Y as X = F(Y)
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Q02. Inverse of Y = a*x^2 + b*x + c is X = a*y^2 + b*y + c
Find the inverse
- Solve for x and we have x = (-b + Sqr(b^2-4*a*(c-y))/(2*a)
- The inverse is y = (-b + Sqr(b^2-4*a*(c-x))/(2*a)
- The inverse is y = (-b - Sqr(b^2-4*a*(c-x))/(2*a)
- The inverse is x = a*y^2 + b*y + c
- 2*a*y + b = Sqr(b^2 - 4*a*(c-x))
- Square both sides : (2*a*y + b)^2 = b^2 - 4*a*(c-x)
- 4*a^2*y^2 + 4*a*b*y + b^2) = b^2 - 4*a*c + 4*a*x
- 4*a*x = 4*a^2*y^2 +4*a*b*y + 4*a*c
- x = a*y^2 + b*y + c
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Q03 Find intersection of y = a*x^2 + B*x + c with it inverse
How many intersctions of y = a*x^2+b*x+c with its inverse
- There is no intersection
- There is one intersection
- There are two intersections
- There are four intersections
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* Find intersection of y = a*x^2 + b*x + c with inverse
* First we solve y = a*x^2 + b*x + c and y = x
* This gives a quadratic equation : a*x^2 + (b-1)*x + c = 0
* Let Discriminant D = (b-1)^2 - 4*a*c
* D < 0 then there is no intersection
* D = 0 there is one intersection, i.e. y = x is tangent to curve
* D > 0 there are 2 or 4 intersections
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Q04 Find intersection of y = x^2 - 2*x + 4 with it inverse
Solution : There is no point of intersection
- Since (b - 1)^2 - 4*a*c = (-2 - 1)^2 - 4*1*4 = -7
- Hence there is no point of intersection
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Graphic Calculater Section 6 10
- Start the progam
- Select section 6 in upper box
- Select 10 in lower box
- Give coefficients 1, -2, 4
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Q05 Find intersection of y = x^2 - 3*x + 4 with it inverse
Funtion y = x^2 - 3*x + 4 and its inverse has 0ne intersections
- Intersection of y = x^2 - 3*x + 4 and y = x
- We solve x^2 - 3*x + 4 = x
- Since D = (b - 1)^2 - 4*a*c = (-3 - 1)^2 - 4*1*4 = 0
- Hence it has only one point of intersection
- The solution is obtained from x^2 - 4*x + 4 = 0
- x = 2
- y = (2)^2 - 3*(2) + 4 = 2
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Graphic Calculater Section 6 10
- Start the progam
- Select section 6 in upper box
- Select 10 in lower box
- Give coefficients 1, -3, 4
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Q05 Find intersection of y = x^2 - 2*x + 4 with it inverse
Solution : There is no point of intersection
- Since (b - 1)^2 - 4*a*c = (-2 - 1)^2 - 4*1*4 = -7
- Hence there is no point of intersection
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Graphic Calculater Section 6 10
- Start the progam
- Select section 6 in upper box
- Select 10 in lower box
- Give coefficients 1, -2, 4
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Q06 Diagram of intersection of y = x^2 - 6*x + 8 with it inverse
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y = x^2 - 6*x + 8 and its inverse 
y = x^2 + 0.25 and its inverse 
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