g(x) = 4x/3-x f(x) = 2x – 5
Given that x > 3, find the exact value of x such that g–1(x) = f(x).
Can you give me a step by step explanation please
Answers (2)
First, it's g^-1(x) which means the inverse function, which means to solve the given function for x.
g(x) = (4x/3) - x Adding parentheses to avoid confusion. This resolves to g(x) = x/3 which is kinda trivial. That's why we use parentheses, and it is not nice to expect someone else to guess what you meant.
g(x) = 4x/(3 - x) Let's go with this guess. The rule is you can do any valid operation on both sides of an equation and it will still be equal. Multiply by (3 - x)/g(x).
3 - x = 4x/g(x) Divide by x.
3/x - 1 = 4/g(x) add 1.
3/x = 4/g(x) + 1 Invert.
x = 3/(4/g(x) + 1)
This is getting to be quite a mess, and I wonder what this is supposed to prove. Math is valid only when it describes reality, and I am not aware of any reality described by this mess.
A function f: R → R is bijective if and only if its graph meets every horizontal and vertical line exactly once.
Stated in concise mathematical notation, a function f with domain X, ( f: X → Y ), is bijective if and only if it satisfies the condition:
for every y in Y there is a unique x in X with y = f(x).
Functions that have inverse functions are said to be invertible. A function is invertible if and only if it is a bijection.
(source: en.wikipedia.org/wiki/Bijection)
g (x) = 4x / (3 - x) ;; we rewrite this
y = 4x / (3-x) ;; and solve for x
y (x - 3) = 4x
3y - xy = 4x
3y = 4x + xy
3y = x (4 + y)
x = 3y / (y + 4)
Now we swap x,y
y = 3x / ( x + 4 ) , which is
g^-1 (x) = 3x / ( x + 4 ) , with
g (x) = 4x / (3 - x) and
g^-1 ( g (x)) = x, D =: {R \ -4, 3}
which essentially tells you that the output of g(x), passed on to g^-1(x) will render the input of g(x). Example:
g (4) = 16 / -1 = -16
and
g^-1 (-16) = -48 / -12 = 4
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Given that x > 3, find the exact value of x such that g^–1(x) = f(x).
with
f(x) = 2x – 5 and
g^-1 (x) = 3x / ( x + 4 ).
Let's do this. Setting these two functions equal gives us
2x – 5 = 3x / ( x + 4 ) ;; solve for x
( 2x – 5 ) ( x + 4 ) = 3x
2x^2 + 8x - 5x - 20 = 3x
2x^2 - 20 = 0
x^2 = 10
As we're told that x > 3 , we don't care about -sqrt (10) and get the exact value
x = sqrt (10) .