Function: For f(x)=2x+3, find a, b so f(ax+b)=x is true for

[TangoFoxtrotGolf]

Let f(x) = 2x + 3

Find values for a and b such that the equation f(ax + b) = x is true for all values of x.

2(ax + b) + 3 = x

So, how does one solve for three variables without some method of substitution?

Or, have I missed the substitution possibilities?

 

[Mrspi]

Re: Function Problem

Let f(x) = 2x + 3

Find values for a and b such that the equation f(ax + b) = x is true for all values of x.

2(ax + b) + 3 = x

So, how does one solve for three variables without some method of substitution?

Or, have I missed the substitution possibilities?

2ax + 2b + 3 = x

You can equate coefficients on the left side and the right side....

(2a)x + (2b + 3) = x

In order for this to be true, the coefficient of x on the left side must be equal to the coefficient of x on the right side. So,

2a = 1

You can solve that for "a".

And the constant term on the left side must equal the constant term on the right side. So,

2b + 3 = 0

And you can solve that for "b".

Once you have values for "a" and "b", check to see that f(2ax + b) = x is true.