G(x)= -1/2x+5

a) use the definition of a one-to-one function, to show that g(x) defines a one-to-one function.

b) find the inverse of g(x)

Given we can conclude the following things:

A) is one-to-one.

B) 's inverse is given by .

Step-by-step explanation:

A) If it is one-to-one, then f(a)=f(b) implies only a=b.

The function is .

Let's see what implies here.

Subtract 5 on both sides:

Multiply both sides by -2:

This is exactly what we need to show that is one-to-one.

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If you look at a visual of on a graph. You would see it passes the horizontal line test making it one-to-one. (It is is just a diagonal line after all.)

B)

The inverse is the swapping of and and then we want to remake the new the subject of the equation.

That is we have to solve the following for :

Subtract 5 on both sides:

Multiply both sides by -2:

So .

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Let's verify.

So we should get that .

Let's see what happens:

-So that looks good so far.

-So that looks good.

Both ways check out so we indeed found the inverse of .