F(x)=2x+1 and g(x)= x^2-4x+3

Question

find the domain of g(x)/f(x) and f(x)/g(x)?

Answer 1

The functions f (x) = 2x + 1 and g (x) = x² - 4x + 3.

• g (x) / f (x) = (x² - 4x + 3)/(2x + 1).

(x² - 4x + 3)/(2x + 1), is a rational function.

We know that all possible values of x  is domain of a function.

A rational function is simply a fraction and in a fraction the denominator cannot be equal to 0 because it would be undefined.

To find which number make the fraction undefined create an equation where the denominator is not equal to zero.

2x + 1 ≠ 0

2x ≠ - 1

x ≠ - 1/2.

The domain of the function : g (x) / f (x) is { x ∈ R : x ≠ - 1/2 }, where R is the set of real numbers.

• f (x) / g (x) = (2x + 1) / (x² - 4x + 3).

(2x + 1) / (x² - 4x + 3), is a rational function.

We know that all possible values of x  is domain of a function.

A rational function is simply a fraction and in a fraction the denominator cannot be equal to 0 because it would be undefined.

To find which number make the fraction undefined create an equation where the denominator is not equal to zero.

x² - 4x + 3 ≠ 0

x² - 3x - x + 3 ≠ 0

x(x - 3) - 1(x - 3) ≠ 0

(x - 3)(x - 1) ≠ 0

Apply  zero product property.

x - 3 ≠ 0  and  x - 1 ≠ 0

x  ≠ 3  and  x ≠ 1.

The domain of the function : f (x) / g (x) is { x ∈ R : x ≠ 1 and x ≠ 3 }, where R is the set of real numbers.