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February 6, 2016
Posted by **Rachal** on Thursday, February 3, 2011 at 4:38am.

Find f^-1, the inverse of f. Then, give the domain and range of f^-1 using interval notation.

f^-1(x)=

Domain (f^-1)=

Range (f^-1)=

Any help is greatly appreciated.

Algebra - helper, Wednesday, February 2, 2011 at 7:05pm

f(x)=(4x-1)/(x+7)

y = (4x-1)/(x+7)

Rewrite as:

y = (4x)/(x+7)- 1/(x+7)

Multiply both sides by x+7:

(x + 7)y = 4x - 1

Expand out terms of the left hand side:

xy + 7y = 4x - 1

xy - 4x = -7y - 1

x(y - 4) = -7y - 1

Divide both sides by y - 4:

x = (-7y - 1)/(y - 4)

f^-1 = (-7x - 1)/(x - 4)

Can you do the domain and range now?

Algebra - Rachal, Wednesday, February 2, 2011 at 7:11pm

I don't know if this is right but this is what I came up with.

f^-1=(-7x+1)/(x-4)

domain f(^-1)=(-inf,-7)U(-7,inf)

range f(^-1)=(-inf,4)U(4,inf)

Let me know if it looks right. Thanks

Math - inverse - MathMate, Thursday, February 3, 2011 at 12:11am

The domain and range suggested apply to f(x). You will see that the vertical asymptote is at x=-7 when the denominator becomes zero.

f(x)=(4x-1)/(x+7)

domain f(^-1)=(-inf,-7)U(-7,inf)

range f(^-1)=(-inf,4)U(4,inf)

The domain and range of f-1(x) is equal to the range and domain respectively of f(x). Double check by evaluating the denominator at the singular points.

Post again if you need confirmation.

I'm still not getting it

- Algebra-I'm still stuck -
**Reiny**, Thursday, February 3, 2011 at 7:16amoriginal:

y = (4x-1)/(x+7)

inverse: interchange x and y variables

x = (4y-1)/(y+7)

cross-multiply

xy + 7x = 4y - 1

xy - 4y = -7x - 1

y(x-4) = -7x - 1

y = (7x+1)/(4-x)

domain: any real x , except x?4

- Algebra-I'm still stuck -
**MathMate**, Thursday, February 3, 2011 at 8:21amYou have corrected stated:

f(x)=(4x-1)/(x+7)

domain f(^-1)=(-inf,-7)U(-7,inf)

range f(^-1)=(-inf,4)U(4,inf)

and Reiny had worked out the inverse:

f^{-1}= (7x+1)/(4-x)

The numerator is a polynomial, so domain is ℝ (always true for polynomials), but denominator is (4-x), which becomes zero and creates a vertical asymptote at x=4.

The domain is the combination of the two, with all the limitations, so it is ℝ-{4}, or in interval notation, (-∞4)∪(4,∞).

Since there is a horizontal asymptote at y=-7, so the range is limited to (-∞7)∪(7,∞).

You will note that the domain of f(x) is the same as the range of f^{-1}(x), and the range of f(x) is the same as the domain of f^{-1}(x). This property is generally true for inverse functions, if an inverse exists.

If you study the graph of f(x) and f^{-1}(x), it will be much easier to understand.

See:

http://img834.imageshack.us/img834/7039/1296725887.png

- Algebra-I'm still stuck -
**Rachal**, Thursday, February 3, 2011 at 6:33pmI want to thank all of you for taking the time to help me understand the problem