SOLUTION: Determine if given function is one to one. If it is, find the formula for the inverse. f(x)=5x-6

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Question 169816:
Determine if given function is one to one. If it is, find the formula for the inverse.
f(x)=5x-6
Found 3 solutions by stanbon, jim_thompson5910, gonzo:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
Determine if given function is one to one. If it is, find the formula for the inverse.
f(x)=5x-6
It is one-to-one because each x-value has one corresponding y-value and
vice versa.
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Inverse:
Interchange x and y to get:
x = 5y-6
Solve for "Y" to get the inverse:
y = (1/5)x + (6/5)
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Cheers,
Stan H.

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
In order to show that f%28x%29 is one-to-one, we need to show that if f%28x%5B1%5D%29=f%28x%5B2%5D%29, then x%5B1%5D=x%5B2%5D.

In other words, we need to show that each y value ONLY has ONE unique x value mapping to it. In this case, we're assuming that x%5B1%5D and x%5B2%5D are two different values that map to the same y value (but as you'll see, they are in fact the same value)



f%28x%5B1%5D%29=f%28x%5B2%5D%29 Start with the given equation


5x%5B1%5D-6=5x%5B2%5D-6 Plug in the function


5x%5B1%5D=5x%5B2%5D Add 6 to both sides. Notice how the "6"s cancel


x%5B1%5D=x%5B2%5D Divide both sides by 5. Notice how the "5"s cancel


So we've shown that if f%28x%5B1%5D%29=f%28x%5B2%5D%29, then x%5B1%5D=x%5B2%5D. This means that f%28x%29 is one-to-one.


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Finding the Inverse:


f%28x%29=5x-6 Start with the given function


y=5x-6 Replace f(x) with y


x=5y-6 Switch x and y


x%2B6=5y Add 6 to both sides


5y=x%2B6 Rearrange the equation


y=%28x%2B6%29%2F5 Divide both sides by 5 to isolate y


So the answer is y=%28x%2B6%29%2F5 which means that the inverse function is

Answer by gonzo(654) About Me  (Show Source):
You can put this solution on YOUR website!
since for every value of x, you get one unique f(x), the equation is a function.
to solve for the inverse do the following:
let y = f(x)
then y = 5x - 6
substitute x for y and y for x.
equation becomes:
x = 5y - 6
solve for y:
add 6 to both sides of equation:
x + 6 = 5y
divide both sides of equation by 5:
(x+6)/5 = y
let y = f%5E-1(x)
you get f%5E-1(x) = (x+6)/5 which is the inverse function of f(x) = 5x-6
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to prove this is an inverse equation, solve for a known value of x in f(x).
f(5) = 5*5-6 = 25-6 = 19
the solution set for f(5) is (5,19)
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take the inverse function of f%5E-1(x) = (x+6)/5
and solve for x = 19 (this is f(x)).
your inverse function starts off as:
f%5E-1(x) = (x+6)/5
you substitute f(x) for x.
since f(x) = 19 which you just solved for when x = 5, then your inverse equation becomes:
f%5E-1(19) = (19+6)/5
the right side of this equation becomes:
(19+6)/5 = 25/5 = 5
your solution set for f%5E-1(19) is (19,5)
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f(x) solution set is (5,19)
f%5E-1(f(x)) solution set is (19,5)
since the x value in the original equation equals the y value in the inverse function equation, this is good.
since the y value in the original equation equals the x value in the inverse function equation, this is also good.
it means that the inverse function was calculated correctly.
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note:
f%5E-1 means the inverse function of f(x)
it does not mean f to the -1 power.
i didn't know how to show it so i used the exponent notation.
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