Question 169816
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^-1}}}(x)
you get {{{f^-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^-1}}}(x) = (x+6)/5
and solve for x = 19 (this is f(x)).
your inverse function starts off as:
{{{f^-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^-1}}}(19) = (19+6)/5
the right side of this equation becomes:
(19+6)/5 = 25/5 = 5
your solution set for {{{f^-1}}}(19) is (19,5)
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f(x) solution set is (5,19)
{{{f^-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^-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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