Find the inverse function of f?

Question

1.) Find the inverse function of f

f(x)=x^2-6x , x greater then or equal to 3

( sorry I don't know how to make that symbol with the keyboard)
(please do it step by step if you can...)

Answer 1

y=x^2 - 6x
=x^2 -2*3x + 3^2 - 3^2
= (x-3)^2 - 9
sq rt. (y+9)=x-3
therfore x= [sq rt.(y+9)] + 3 whre x>=3 [inverse func.]

Answer 2

f(x) = x² - 6x , x greater than or equal to 3.

y = x² - 6x
y + 9 = x² - 6x + 9
y + 9 = (x - 3)²
y = (x - 3)² - 9, where x ≤ 3

This is the right half of a parabola.

The inverse function will be the top half of a parabola that opens to the right. To find it reverse x and y and solve for y.

x + 9 = (y - 3)²
√(x + 9) = y - 3
√(x + 9) + 3 = y
y = √(x + 9) + 3

The inverse function is:

f‾¹(x) = √(x + 9) + 3

Answer 3

f(x) = x² - 6x, x ≥ 3

Remember that when taking the inverse, the domain becomes the range and the range becomes the domain. We should first obtain the range of this function. We do so by finding the vertex, and putting this in vertex form.

f(x) = x² - 6x + 9 - 9
f(x) = (x - 3)² - 9

Implying there is a vertex at (3, -9), and telling us the range of this function is y ≥ 9, for y = f(x)

To find the inverse, make f(x) = y and take advantage of the
vertex form.

y = (x - 3)² - 9.

Swap the x and y variables.

x = (y - 3)² - 9

Solve for y.

x + 9 = (y - 3)²

Take the square root of both sides.

± √(x + 9) = y - 3

Note though, that the range of f(x) was y ≥ -9, which means the domain will be x ≥ -9. Also, the domain of f(x) was
x ≥ 3, so the range of this will be y >= 3.

For that reason, we discard the negative possibility, and

√(x + 9) = y - 3

Isolate y.

y = √(x + 9) + 3

And make your concluding remark.

fˉ¹(x) = √(x + 9) + 3

Answer 4

f(x) = x^2 - 6x ; x>= 3
y = x^2 - 6x
Range is
x>=3
x^2 >=9 and 6x >= 18
x^2 - 6x >= -9 or y >= -9



funtion inverse is
x = y^2 - 6y
y^2 - 6y - x =0
y = (6 (+-) square-root(36 + 4x) ) / 2
y = 3 (+-) square-root(9 + x)
f-1(x) = 3 + square-root(9 + x)
f-1(x) = 3 - square-root(9 + x)

Domain f-1 is Range of f
that mean x >=-9

Answer 5

y = x^2-6x => x^2-6x -y = 0

or x = [6 +sqrtr(36+4y)]/2 or x = [6 - sqrtr(36+4y)]/2
we know that x >= 3 ; therefore y >= -9
Taking only one of the roots :
inverse is = [3 +sqrtr(9+x)] for x >= -9

Answer 6

y=x^2-6x , x>=3
at x=3, y=-9

therefore,
inverse is x=y^2-6y, y>=-9

that's it, since inverses are just a role reversal in crude terms.

Answer 7

f(x)=x^2-6x
y=x^2-6x
x=y^2-6x
therefore y=2-6xsqx
so the answer is x^x-3

Answer 8

E = MC squared