this exercise involves a difference quotient for an exponential function. if f(x)=3^x - 1, show that…

this exercise involves a difference quotient for an exponential function. if f(x)=3^x - 1, show that (\frac{f(x + h)-f(x)}{h}=(3^x - 1)\frac{3^h - 1}{h}). simplify your answers completely at each step. f(x)=3^x - 1 so (\frac{f(x + h)-f(x)}{h}=\frac{(3^{x + h}-1)-(3^x - 1)}{h}=\frac{3^{x + h}-1 - 3^x + 1}{h}=\frac{3^xcdot3^h-3^x}{h}=(3^x - 1)\frac{3^h - 1}{h})

this exercise involves a difference quotient for an exponential function. if f(x)=3^x - 1, show that (\frac{f(x + h)-f(x)}{h}=(3^x - 1)\frac{3^h - 1}{h}). simplify your answers completely at each step. f(x)=3^x - 1 so (\frac{f(x + h)-f(x)}{h}=\frac{(3^{x + h}-1)-(3^x - 1)}{h}=\frac{3^{x + h}-1 - 3^x + 1}{h}=\frac{3^xcdot3^h-3^x}{h}=(3^x - 1)\frac{3^h - 1}{h})

Answer

Explanation:

Step1: Find (f(x + h))

Given (f(x)=3^{x}-1), then (f(x + h)=3^{x + h}-1).

Step2: Calculate (f(x + h)-f(x))

[ \begin{align*} f(x + h)-f(x)&=(3^{x + h}-1)-(3^{x}-1)\ &=3^{x + h}-1 - 3^{x}+1\ &=3^{x}\cdot3^{h}-3^{x}\ &=3^{x}(3^{h}-1) \end{align*} ]

Step3: Calculate (\frac{f(x + h)-f(x)}{h})

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

Answer:

(3^{x}\left(\frac{3^{h}-1}{h}\right))