Show that the given function is one-to-one and find its inverse.

Mathematics
Tutor: None Selected Time limit: 1 Day

Oct 2nd, 2015

Hello!

The function f has the domain (-∞, +∞) and the same range.

To find the inverse (denote it as g) we have to solve the equation

f(g(x)) = x.

f(g(x)) = 8*g(x) + 10 = x,

g(x) = (x-10)/8.

The function g also has the domain and range (-∞, +∞).
Actually, having inverse function is sufficient to be a one-to-one.


Checking algebraically:

f(g(x)) = 8*[(x-10)/8] + 10 = (x-10) + 10 = x,
g(f(x)) = [f(x)-10]/8 = [8x+10 - 10]/8 = 8x/8 = x.

For the graphs, see https://www.desmos.com/calculator/dw9iejncaw
The graphs of f and g are symmetric with respect to y=x.

Please ask if something is unclear.
Oct 2nd, 2015

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