What did you get for f-1(x)?
And what did you get for f-1(f(x))?
How would you do the inverse of this function: -1/7sqrt(16-x^2)?
-I know you switch the x and y and solve for y.
I keep on getting a weird answer. Help!
3 answers
In general, there are different ways you can verify your answer, and in this case, first verify the question!
For a function f(x) to have an inverse, it must be one-to-one and onto.
A function must pass the vertical line test (i.e. any vertical line within the domain must not have two or more values for the image).
A one-to-one function must pass the vertical line test and the horizontal line test (i.e. any horizontal line within the range must not have two or more values in the domain).
The given function does not pass the horizontal line test because it is an even function, i.e. for any value of x on its domain (-4,4), f(x)=f(-x).
Before finding the inverse, we must restrict the domain of the given function to either [0,4) or (-4,0]. We will choose [0,4). The other option can be treated similarly.
If you work out the math, which I am sure you did, after the solution of the equations, the inverse turns out to be either
g(x)=-sqrt(784*x^2-1)/(7*x) or
g(x)=sqrt(784*x^2-1)/(7*x)
Which one do we choose?
The answer lies in the domain that we have chosen in the first place.
We have chosen the domain to be [0,4), for which the image is always negative. Thus the image of the inverse g(x) has to be [0,4), and its domain has to be negative.
For g(x) to be positive when the domain is negative, we can only choose the first option, or
g(x)=-sqrt(784*x^2-1)/(7*x)
A graphics plot will make this all clear:
http://imageshack.us/photo/my-images/43/1312958033.png/
For a function f(x) to have an inverse, it must be one-to-one and onto.
A function must pass the vertical line test (i.e. any vertical line within the domain must not have two or more values for the image).
A one-to-one function must pass the vertical line test and the horizontal line test (i.e. any horizontal line within the range must not have two or more values in the domain).
The given function does not pass the horizontal line test because it is an even function, i.e. for any value of x on its domain (-4,4), f(x)=f(-x).
Before finding the inverse, we must restrict the domain of the given function to either [0,4) or (-4,0]. We will choose [0,4). The other option can be treated similarly.
If you work out the math, which I am sure you did, after the solution of the equations, the inverse turns out to be either
g(x)=-sqrt(784*x^2-1)/(7*x) or
g(x)=sqrt(784*x^2-1)/(7*x)
Which one do we choose?
The answer lies in the domain that we have chosen in the first place.
We have chosen the domain to be [0,4), for which the image is always negative. Thus the image of the inverse g(x) has to be [0,4), and its domain has to be negative.
For g(x) to be positive when the domain is negative, we can only choose the first option, or
g(x)=-sqrt(784*x^2-1)/(7*x)
A graphics plot will make this all clear:
http://imageshack.us/photo/my-images/43/1312958033.png/
On re-reading the question, I just realized from the way you have punctuated the question, f(x) could be either
f(x)=-1/7sqrt(16-x^2)
or simply
f(x)=1/7sqrt(16-x^2).
In the above response, I have taken it to be
f(x)=-1/7sqrt(16-x^2)
If f(x) has the opposite sign, it will be reflected about the x-axis, and the inverse about the y-axis.
It would make a good exercise for your arguments in this case.
f(x)=-1/7sqrt(16-x^2)
or simply
f(x)=1/7sqrt(16-x^2).
In the above response, I have taken it to be
f(x)=-1/7sqrt(16-x^2)
If f(x) has the opposite sign, it will be reflected about the x-axis, and the inverse about the y-axis.
It would make a good exercise for your arguments in this case.
1 answer