Hannah H.
asked • 12/30/16Function problem
F(x)=4x^2-16
how can the domain of function f be restricted so that f^-1 is also a function?
what is the equation of f^-1?
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2 Answers By Expert Tutors
Inactive Tutor answered • 12/30/16
Tutor
New to Wyzant
Kenneth gave you the graphical/analytical solution. I will give the algebraic solution.
Let F(x)=y
Switch places with x and y.
x = 4y2 - 16
Solve for y.
x + 16 = 4y2
±√(x + 16) = 2y
±(1/2)√(x + 16) = y ----> sideways parabola
Set x + 16 ≥ 0
x ≥ -16
The domain of f-1 is all real numbers greater than or equal to -16.
Stephen M.
tutor
If you leave the ± in, then it isn't a function (fails vertical line test). And you gave the domain for f-1(x), not f(x). I believe she was looking for a restriction on the domain of the initial function. What she's looking for in that case is x ≥ 0, although x ≤ 0 would also work. For that matter, x > 2 would also work just as well.
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12/30/16
Inactive Tutor
The question asked how to restrict the domain of f(x). The domain of f(x) is all real numbers. But when you get its inverse, the domain becomes limited. Also, inverse functions must pass the horizontal line test, not the vertical line test.
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12/30/16
Stephen M.
tutor
You're right that the domain of f(x) as written can be all real numbers. If the domain of f(x) is all real numbers, then f-1(x) isn't a function. A function must pass the vertical line test to be a function. The horizontal line test applied to a function tells you whether its inverse is a function. Once you've solved for the inverse, the vertical line test tells you whether your solution is also a function.
The domain of f(x) can be arbitrarily restricted to any segment of the real numbers. The question is asking for a restriction of the domain that results in the inverse being a function. If we define f(x) = 4x2 - 16; x ≥ 0, we have restricted the domain of our function so that its inverse is also a function. For the record, Kenneth's initial answer was correct.
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12/30/16
Inactive Tutor
"For the record, Kenneth's initial answer was correct". That statement is insulting to the other tutor and undermines his effort to help the student answer the question.
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12/31/16
Inactive Tutor answered • 12/30/16
Tutor
New to Wyzant
The function you have is a parabola(quadratic function). Discover its vertex; restrict x values to be greater than or equal to that vertex's x value (which is zero, in this case). Now your function is the upward-rising part of the original function only, and this graph will pass the Horizontal Line Test, of course, and therefore has an inverse.
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Inactive Tutor
12/30/16