Range

Find the range of the function f(x) = 9^x - 3^x + 1.

Hello Guest!

The range is the space between the minimum of the function and the maximum value.

$f(x) = 9^x - 3^x + 1\\ \frac{df(x)}{dx}=9^xln\ 9-3^xln\ 3=0\\ 9^xln\ 9=3^xln\ 3$

$x_{min}= -0.6309297535714576$

$f(x) = 9^x - 3^x + 1\\ f(x_{min}) = 9^{(-0.6309297535714576)} - 3^{(-0.6309297535714576)} + 1$

$f(x_{min})=0.75$

$\color{black}The\ range\ is\\ f(x)\in \mathbb R\ |\ 0.75\leq y<\infty$

  !