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A Level maths core 3

by Sarah
(London)










































The function f is defined by f(x) = sq root (mx+7) - 4, where x is equal to or greater than -7/m and m is a positive constant.


(iii) y=f(x) and y =f^-1 (x) do not meet. Explain how it can be deduced that neither curve meets the line y=x, and hence determine the set of possible values of m.

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Nov 22, 2011
Function and Inverse Function
by: Staff


Question:

by Sarah
(London)

The function f is defined by f(x) = sq root (mx+7) - 4, where x is equal to or greater than -7/m and m is a positive constant.


(iii) y=f(x) and y =f^-1 (x) do not meet. Explain how it can be deduced that neither curve meets the line y=x, and hence determine the set of possible values of m.



Answer:

As I’m sure you already know, an inverse function is a (symmetric) reflection of a function across the y = x axis.

However, a function and its inverse can not only touch the y = x axis, they can cross back and forth multiple times (it depends on the function, of course). When a function crosses the y = x axis, its inverse will also cross the y=x axis at exactly the same point. Both the function and the inverse function will be equal to one another when this occurs.

Will this happen for the function in your problem: f(x) = sq root (mx+7) - 4 ?

To find out, just set the function f(x) = x, and solve for x.

f(x) = sqrt(mx+7) - 4 = x

sqrt(mx+7) - 4 = x

sqrt(mx+7) - 4 + 4 = x + 4

sqrt(mx+7) + 0 = x + 4

sqrt(mx+7) = x + 4

[sqrt(mx+7)]² = (x + 4)²

mx + 7 = x² + 8x + 16

mx - mx + 7 - 7 = x² + 8x + 16 - mx - 7

0 + 0 = x² + 8x + 16 - mx - 7

0 = x² + 8x + 16 - mx - 7

x² + 8x + 16 - mx - 7 = 0

x² + 8x - mx + 16 - 7 = 0

x² + (8 – m)x + 9 = 0


since this is an ordinary quadratic equation, we can use the quadratic formula to solve for x
ax² + bx + c = 0

x = [-b ± √(b² - 4ac)]/(2a)


x² + (8 - m)x + 9 = 0

x = {-(8 - m) ± √[(8 - m)² - 4*1*9]}/(2*1)

x = {-(8 - m) ± √[(8 - m)² - 4*1*9]}/2

x = -4 + m/2 ± {√[(8 - m)² - 4*1*9]}/2

x = -4 + m/2 ± {√[(8 - m)² - 36]}/2

x = -4 + m/2 ± {√[(8 - m)² - 36]}/2


if this solution is valid for a particular value of m, that means the function and its inverse touch the y=x axis at that point

to have a solution for any particular value of m, the square root of the discriminant (the expression under the square root sign) must not include a complex number.

We are looking for values of m that don’t provide a solution. This means that the discriminant must be less than 0.


(8 - m)² - 36 < 0

When 2 ≤ m ≤ 14 the discriminant is a negative number. This means the function and its inverse do not touch the y=x axis when 2 ≤ m ≤ 14.


The final answer to your question is:

y=f(x) and y =f^-1 (x) do not touch the y=x axis when 2 ≤ m ≤ 14.



Thanks for writing.

Staff
www.solving-math-problems.com



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