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# Range

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

Aug 11, 2021

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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.

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

$$x_{min}= -0.6309297535714576$$

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

$$f(x_{min})=0.75$$

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

!

Aug 12, 2021
edited by asinus  Aug 12, 2021