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Question 856270: Find the inverse of the one-to-one function.
f(x)=5x-3/2
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! start with y = 5x - 3/2
interchange y and x to get:
x = 5y - 3/2
solve for y as follows:
add 3/2 to both sides of the equation to get:
x + 3/2 = 5y
commute this equation to get:
5y = x + 3/2
divide both sides of this equation by 5 to get:
y = x/5 + 3/10
that should be your inverse function.
if it is, then f(g(x)) should be equal to x and g(f(x)) should be equal to x.
make your original equation f(x).
make the inverse equation equal to g(x).
you get:
f(x)= 5x - 3/2
g(x) = x/5 + 3/10
f(g(x)) is derived by replacing x in f(x) = 5x - 3/2 with (x/5 + 3/10) from g(x) to get:
f(g(x)) = 5*(x/5 + 3/10) - 3/2
simplify by distributing the multiplication to get:
f(g(x)) = 5 * x/5 + 5 * 3/10 - 3/2
simplify to get:
f(g(x)) = x + 15/10 - 3/2
divide numerator and denominator of 15/10 by 5 to get:
f(g(x)) = x + 3/2 - 3/2
combine like terms to get:
f(g(x)) = x
one half of the test is done.
now to the other half.
g(f(x)) is derived by replacing x in g(x) = x/5 + 3/10 with (5x - 3/2) from f(x) to get:
g(f(x)) = (5x - 3/2) / 5 + 3/10
multiply (5x - 3/2) / 5 * (2/2) to get:
g(f(x)) = 2*(5x-3/2)/10 + 3/10
combine the numerators under the common denominator to get:
g(f(x)) = (2*(5x-3/2) + 3) / 10
simplify by distributing the multiplication in the numerator to get:
g(f(x)) = (10x - 3 + 3) / 10
simplify by combining like terms to get:
g(f(x)) = 10x / 10
the 10 in the numerator and the 10 in the denominator cancel out and you are left with:
g(f(x)) = x
the test for inverse functions is passed, so your solution is:
f(x) = 5x - 3/2
g(x) = x/5 + 3/10 = inverse function of f(x)
the graphs of inverse functions are symmetric about the line y = x.
the graph of these 2 functions is shown below:
they look symmetric.
another test is that f(x,y) is equal to g(y,x)
the terminology is scary and difficult to follow, but this is what happens:
start with:
f(x) = 5x - 3/2 and replace f(x) with y to get:
y = 5x - 3/2
pick any value of x and solve for y.
i picked x = 5 to get:
y = 5(5) - 3/2 = 25 - 3/2 = 50/2 - 3/2 = 47/2.
this results in:
y = 47/2
now look at g(x) = x/5 + 3/10 and replace g(x) with y to get:
y = x/5 + 3/10
now replace x in this equation with y from the first equation to get:
y = (47/2)/5 + 3/10
simplify this to get:
y = 47/10 + 3/10
combine the numerators under the common denominator of 10 to get:
y = 50/10 which is equal to 5.
you get y = 5
f(x,y) is equal to (5,47/2)
g(x,y) is equal to (47/2,5)
the x from g(x) is equal to the y from f(x).
the y from g(x) is equal to the x from f(x).
this is what they mean when they say f(x,y) = g(y,x).
you take the output from f(x) and make it the input to g(x) and the result is you are back to x.
5 becomes 47/2 after being put through f(x) and then 47/2 becomes 5 after being put through g(x).
the inverse function undoes what the original function does.
f(x) makes 5 become 47/2 and g(x) undoes that by making 47/2 become 5.
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