MHB Inverse Functions & Newton's Method: Debmnzl's Question | Yahoo! Answers

AI Thread Summary
The discussion focuses on finding the inverse of the function F(x) = a loge(x) + 1/2, which is determined to be F^(-1)(x) = e^(2x - 1)/(2a). The user is tasked with finding a specific value of 'a' such that the graphs of F(x) and its inverse touch at exactly one point. The solution involves using Newton's method to solve the equation a = e^((2a - 1)/(2a)), leading to an approximate value of a = 2.156. Consequently, the point of intersection for the function and its inverse is approximately (2.156, 2.156).
MarkFL
Gold Member
MHB
Messages
13,284
Reaction score
12
Here is the question:

A math function question please help?

F(x) = a loge(x) + 1/2
Find the inverse which I found is e^x/a

It then said let a=2 and sketch graph which I did.

I'm getting trouble at this part

Find the value of a in 3 decimal places such that the graphs of y=f(x) and its inverse touch each other once only.

And hence find the contact point.

Please help explanations too please

Here is a link to the question:

A math function question please help? - Yahoo! Answers

I have posted a link there to this topic so that the OP may find my response.
 
Mathematics news on Phys.org
Hello Debmnzl,

We are given the function:

$\displaystyle f(x)=a\ln(x)+\frac{1}{2}$

and are asked to compute its inverse. So, we may begin by switching $x$ and $f(x)$, and then solving for $f(x)$ which will actually be $\displaystyle f^{-1}(x)$

$\displaystyle x=a\ln(f(x))+\frac{1}{2}$

Subtract through by $\displaystyle \frac{1}{2}$

$\displaystyle x-\frac{1}{2}=a\ln(f(x))$

Get common denominator on the left:

$\displaystyle \frac{2x-1}{2}=a\ln(f(x))$

Divide through by $a$:

$\displaystyle \frac{2x-1}{2a}=\ln(f(x))$

Convert from logarithmic to exponential form:

$\displaystyle e^{\frac{2x-1}{2a}}=f(x)$

and so we have:

$\displaystyle f^{-1}(x)=e^{\frac{2x-1}{2a}}$

Now, as a means of checking our work, we may use the fact that we require:

$\displaystyle f\left(f^{-1}(x) \right)=f^{-1}(f(x))=x$

i) $\displaystyle f\left(f^{-1}(x) \right)=a\ln\left(e^{\frac{2x-1}{2a}} \right)+\frac{1}{2}=a\left(\frac{2x-1}{2a} \right)+\frac{1}{2}=x$

ii) $\displaystyle f^{-1}(f(x))=e^{\frac{2\left(a\ln(x)+\frac{1}{2} \right)-1}{2a}}=e^{\frac{2a\ln(x)}{2a}}=e^{\ln(x)}=x$

So, we know the inverse function we found is correct.

Next, we are instructed to let $a=2$ and sketch the graph. I will assume we are to graph both the given function and its inverse:

View attachment 613

Now, to find a value of $a$ such that the function and its inverse touch only once, we may use the fact that the two are reflected about the line $y=x$, and so must be tangent to this line.

The gradient of the line $y=x$ is 1, and so, we require:

$\displaystyle \frac{d}{dx}\left(f(x) \right)=\frac{a}{x}=1\,\therefore\,a=x$

$\displaystyle \frac{d}{dx}\left(f^{-1}(x) \right)=\frac{e^{\frac{2x-1}{2a}}}{a}=1\,\therefore\,a=e^{\frac{2x-1}{2a}}$

Now since we found $a=x$ we may write:

$\displaystyle a=e^{\frac{2a-1}{2a}}$

To compute the solution to this equation to 3 decimal places, we may use Newton's method. Let's define:

$\displaystyle g(a)=e^{\frac{2a-1}{2a}}-a=0$

where:

$\displaystyle g'(a)=\frac{1}{2a^2}e^{\frac{2a-1}{2a}}-1$

Newton's method gives us the recursion:

$\displaystyle a_{n+1}=a_n-\frac{g(a_n)}{g'(a_n)}$

Using the function we defined and its derivative, we have:

$\displaystyle a_{n+1}=a_n-\frac{e^{\frac{2a_n-1}{2a_n}}-a_n}{\frac{1}{2a_n^2}e^{\frac{2a_n-1}{2a_n}}-1}=\frac{a_n(1-2a_n)}{1-2a^2e^{\frac{1-2a_n}{2a_n}}}$

Using $a_0=2$ as our initial estimate, we then find:

$a_1\approx2.15910252177$

$a_2\approx2.15553677378$

$a_3\approx2.1555352035$

$a_4\approx2.1555352035$

Now, since we were only asked to find the estimate to 3 decimal places, we would write:

$a\approx2.156$

and since we found $a=x$ then the $y$-coordinate of the point of intersection must also be $a$ since the function and its inverse will touch along the line $y=x$

And so the point of intersection is approximately (2.156,2.156). Here is a graph using the better approximation we found:

View attachment 614
 

Attachments

  • debmnlz.jpg
    debmnlz.jpg
    6.5 KB · Views: 101
  • debmnlz2.jpg
    debmnlz2.jpg
    7.6 KB · Views: 100
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
Is it possible to arrange six pencils such that each one touches the other five? If so, how? This is an adaption of a Martin Gardner puzzle only I changed it from cigarettes to pencils and left out the clues because PF folks don’t need clues. From the book “My Best Mathematical and Logic Puzzles”. Dover, 1994.
Back
Top