Functions f and g are defined by
f(x)=4x-2k
g(x)=9/(2-x)
i.Find the values of k for which the equation fg(x)=x has two equal roots.
f(g(x))
= f(9/(2-x))
= 4(9/(2-x)) - 2k
so 4(9/(2-x)) - 2k = x
multiply by 2-x
36 - 2k(2-x) = x(2-x)
36 - 4k + 2kx = 2x - x^2
x^2 + x(2k - 2) + 36 - 4k = 0
to have 2 equal roots, the discriminant has to be zero
(2k-2)^2 - 4(1)(36-4k) = 0
4k^2 - 8k + 4 - 144 + 16k = 0
4k^2 + 8k - 140 = 0
k^2 + 2k - 35 = 0
(k+7)(k-5) = 0
k = -7 or k = 5
Well, to find the values of k for which fg(x) has two equal roots, we need to set fg(x) equal to x and solve for k. So let's set up the equation:
fg(x) = x
Substituting the expressions for f(x) and g(x), we get:
(4x - 2k) * (9 / (2 - x)) = x
Now, let's simplify this equation by getting rid of the fraction:
(4x - 2k) * 9 = x * (2 - x)
Expanding both sides, we have:
36x - 18k = 2x^2 - x^2
Combining like terms:
x^2 + 36x - 18k = 0
To find the values of k, we need to find the discriminant of this quadratic equation. The discriminant is given by b^2 - 4ac, where a = 1, b = 36, and c = -18k.
Using the quadratic formula, we know that if the discriminant is equal to zero, then the equation has two equal roots:
Discriminant = b^2 - 4ac
= 36^2 - 4(1)(-18k)
= 1296 + 72k
Setting the discriminant equal to zero:
1296 + 72k = 0
Solving for k:
72k = -1296
k = -18
So, the value of k for which the equation fg(x) = x has two equal roots is k = -18.
To find the values of k for which the equation fg(x) = x has two equal roots, we need to set the equation equal to zero and find the values of k for which this equation has a double root.
The equation fg(x) = x can be expressed as:
f(g(x)) = x
Substituting the given functions:
f(g(x)) = 4(g(x)) - 2k
We can rewrite g(x) as:
g(x) = 9 / (2 - x)
Substituting g(x) back into the equation:
f(g(x)) = 4(9 / (2 - x)) - 2k
Now, we set f(g(x)) equal to x:
4(9 / (2 - x)) - 2k = x
Multiplying both sides of the equation by (2 - x) to eliminate the denominator:
36 - 4x - 2k(2 - x) = x(2 - x)
Expanding and rearranging the equation:
36 - 4x - 4k + 2kx = 2x - x^2
Rearranging to get a quadratic equation:
x^2 - (6 - 2k)x + (36 - 4k) = 0
For this equation to have two equal roots, its discriminant should be equal to zero. The discriminant can be calculated using the formula:
Discriminant = b^2 - 4ac
In this case, a = 1, b = -(6 - 2k), and c = (36 - 4k). We can substitute these values into the discriminant formula:
(6 - 2k)^2 - 4(1)(36 - 4k) = 0
Expanding and simplifying:
36 - 24k + 4k^2 - 144 + 16k = 0
Combining like terms:
4k^2 - 8k - 108 = 0
Now, we can solve this quadratic equation for k.
To find the values of k for which the equation fg(x) = x has two equal roots, we need to set fg(x) equal to x and solve for x.
First, we'll substitute f(x) and g(x) into the equation fg(x) = x:
(4x - 2k) * (9 / (2 - x)) = x
Next, we can simplify the equation by getting rid of the fractions. Multiply both sides of the equation by (2 - x):
(4x - 2k) * (9 / (2 - x)) * (2 - x) = x * (2 - x)
Simplifying further:
(4x - 2k) * 9 = x * (2 - x) * (2 - x)
36x - 18k = x * (2 - x) * (2 - x)
Now, we can expand the right side of the equation:
36x - 18k = x * (4 - 4x + x^2)
36x -18k = 4x - 4x^2 + x^3
Simplifying further:
36x - 18k = x^3 - 4x^2 + 4x
Now, we'll move all the terms to one side of the equation to obtain a cubic equation:
0 = x^3 - 4x^2 + 4x - 36x + 18k
Combining like terms:
0 = x^3 - 4x^2 - 32x + 18k
To find the values of k, we need to solve this cubic equation. While there are different methods to solve cubic equations, it is often lengthy and best solved using numerical methods or software. The method used depends on the availability of such resources and the accuracy required.