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

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)   (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.
---------------
Inverse:
Interchange x and y to get:
x = 5y-6
Solve for "Y" to get the inverse:
y = (1/5)x + (6/5)
==================
Cheers,
Stan H.
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!
In order to show that is one-to-one, we need to show that if , then .

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 and are two different values that map to the same y value (but as you'll see, they are in fact the same value)

Start with the given equation

Plug in the function

Add 6 to both sides. Notice how the "6"s cancel

Divide both sides by 5. Notice how the "5"s cancel

So we've shown that if , then . This means that is one-to-one.

===============================================

Finding the Inverse:

Start with the given function

Replace f(x) with y

Switch x and y

Add 6 to both sides

Rearrange the equation

Divide both sides by 5 to isolate y

So the answer is which means that the inverse function is
Answer by gonzo(654)   (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 = (x)
you get (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 (x) = (x+6)/5
and solve for x = 19 (this is f(x)).
your inverse function starts off as:
(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:
(19) = (19+6)/5
the right side of this equation becomes:
(19+6)/5 = 25/5 = 5
your solution set for (19) is (19,5)
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f(x) solution set is (5,19)
(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:
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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