SOLUTION: Let g(x) = 2x � 5. Is g(x) one-to-one? If it is, find a formula for its inverse

Question 227824: Let g(x) = 2x � 5. Is g(x) one-to-one? If it is, find a formula for its inverse
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
g(x) = 2x - 5 is the equation of a line and is a function because there is only one value of y for each value of x.

The inverse function of g(x) is found by:

Solving for x.
Replacing x with y and y with x.

Let y = g(x)

Your equation becomes y = 2x - 5

Solve for y.

subtract y from both sides of the equation to get:
0 = 2x - 5 - y
subtract 2x from both sides of the equation to get:
-2x = -y - 5
divide both sides of the equation by -2 to get:
x = y/2 + 5/2

Replace x with y and y with x to get:

y = x/2 + 5/2

If this is the inverse function, then both equations will be a reflection about the line y = x.

A graph of these equations and the equation of y = x is shown below:

If these are inverse functions, then:

f(g(x) = g(f(x)

Take f(g(x))

f(x) = 2x-5
g(x) = x/2 + 5/2

f(g(x) = f(x/2+5/2) = 2 * (x/2 + 5/2) - 5 = x + 5 - 5 = x

g(f(x) = g(2x-5) = (2x-5)/2 + 5/2 = x - 5/2 + 5/2 = x

We have f(g(x) = g(f(x) which confirms that these equations are inverse functions of each other.

Definition of a function states that you have a 1 to 1 mapping form x to y.

This happens with both these equations as shown in the graph.

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