What is the range of the function f(x)=2/(3x-1)?

Mar 19, 2017

The range of $f \left(x\right)$ is ${R}_{f \left(x\right)} = \mathbb{R} - \left\{0\right\}$

Explanation:

Let $y = \frac{2}{3 x - 1}$

Then,

$\left(3 x - 1\right) = \frac{2}{y}$

$3 x = \frac{2}{y} + 1 = \frac{2 + y}{y}$

$x = \frac{2 + y}{3 y}$

The inverse function of $f \left(x\right)$ is

${f}^{-} 1 \left(x\right) = \frac{2 + x}{3 x}$

The range of $f \left(x\right)$ is $=$ the domain of ${f}^{-} 1 \left(x\right)$

As we cannot divide by $0$, $x \ne 0$

The domain of ${f}^{-} 1 \left(x\right)$ is ${D}_{{f}^{-} 1 \left(x\right)} = \mathbb{R} - \left\{0\right\}$

The range of $f \left(x\right)$ is ${R}_{f \left(x\right)} = \mathbb{R} - \left\{0\right\}$