Find the roots of the function f(x) = (2^x − 1) · (x^2 + 2x − 3) with x ∈ R

Answer 1

Hello, the roots of a function are where the function crosses the x-axis

in other words where f(x) = 0. Given this we get the following equation

0 = (2^x-1)*(x²+2x-3)

When you first look at this, it can seem like a very tough problem. If we start to think about each expression in parenthesis as just a number then we can simplify the problem as follows. Suppose the equation was written as something like 0 = y*z. For this to be true either y or z has to be 0. You can never multiply two non-zero numbers and end up with 0. This means we can simplify our problem and say the following:

0 = 2^x-1 and 0 = x²+2x-3

First equation:

0 = 2^x-1

2^x= 1 therefore x must be 0

Second equation:

0 = x²+2x-3

we have to factor it or use the quadratic formula

0 = (x+3)(x-1)

x = -3 or 1

Therefore our roots are 0, -3, 1 (if we plug any of these numbers back in the original expression it will equal 0, we can use that to double check that we did not make any calculation mistakes)

If this does not make sense please let me know and I will elaborate.