Good evening to all around the world.
Need help with this question, however i must show my answers supported by the working out. Please help.
This question is difficult, the one who solves it for me is my hero!!!!
Q11)
Two functions f and g are defined by
F:x (arrow) 25/(3x-2)
g:x (arrow) x^2
Find
a) The inverse function f-1
b)The composite function fg
c) The solution of equation fg(x)+2/(x-1).
Please show me how to work it out so i understand. Thank you!!!!! x
What does (arrow) mean?
If we can assume it means =
I think that is right
y = 25/(3x-2)
find inverse
x = 25/(3y-2)
3 x y - 2 x = 25
3 x y = (2x - 25)
y = (2x-25)/3x
f(x^2) = 25/(3x^2-2)
c is not an equation, no = sign
so no solution
Sorry i wrote c wrong
its
the solution of equation fg(x)=2/(x-1)
really sorry, had less sleep last night
25/(3x^2-2) = 2/(x-1)
25(x-1) = 6 x^2 - 4
25 x - 25 = 6 x^2 - 4
6x^2 -25 x +21 = 0
(6x-7)(x-3) = 0
x = 3 or 7/6
Could you ellaborate further on how you got your answer for b please Damon.
The composite function fg
thank you
Good evening! I'll be happy to help you with this question. Let's break it down step by step and work through each part.
a) To find the inverse function f-1, we need to interchange the roles of x and y in the original function f, and solve for y.
The original function f is:
f(x) = 25/(3x-2)
Let's replace f(x) with y:
y = 25/(3x-2)
We can solve this equation for x in terms of y by following these steps:
Step 1: Swap x and y:
x = 25/(3y-2)
Step 2: Solve for y:
3xy - 2x = 25
3xy = 25 + 2x
y = (25 + 2x)/(3x)
So, the inverse function f-1 is:
f-1(x) = (25 + 2x)/(3x)
b) To find the composite function fg, we need to substitute the output of the function g (x^2) into the function f.
The function g is:
g(x) = x^2
Now let's substitute g(x) into f:
f(g(x)) = f(x^2)
Substituting f(x) from the original function:
f(x^2) = 25/(3(x^2) - 2)
Therefore, the composite function fg is:
fg(x) = 25/(3x^2 - 2)
c) To find the solution of the equation fg(x) + 2/(x-1), we need to set the equation equal to zero and solve for x.
The equation is:
fg(x) + 2/(x-1) = 0
Substituting fg(x) with the previously found expression:
25/(3x^2 - 2) + 2/(x-1) = 0
To solve this equation, we can multiply both sides by the common denominator to eliminate the fractions:
[(25 * (x-1))/(3x^2 - 2)] + [2(x-1)] = 0
Simplifying the equation:
[(25x-25)/(3x^2 - 2)] + [2x - 2] = 0
Now, we can combine the fractions by finding the common denominator, which is (3x^2 - 2):
[25x - 25 + 2(x-1)(3x^2 - 2)] / (3x^2 - 2) = 0
Simplifying the numerator and expanding the expression:
[25x - 25 + 6x^3 - 4x - 6x^2 + 4] / (3x^2 - 2) = 0
Combining like terms:
[6x^3 - 6x^2 + 21x - 21] / (3x^2 - 2) = 0
Now, to find the solution, you can either solve this equation algebraically by factoring or using numerical methods such as graphing or using a calculator. The exact solution will depend on the specific values of x that satisfy the equation.
I hope this helps you understand how to approach and solve this question! If you have any further questions, feel free to ask.