Question

These exercises involve a difference quotient for an exponential function.

If $f\left(x\right)=3^{x-1}$, show that

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

Short answer

We have shown that $\frac{f(x+h)-f\left(x\right)}{h}=3^{x-1}\left(\frac{3^h-1}{h}\right)$.

Step-by-step solution

Step 1. Given.

The given function is $f\left(x\right)=3^{x-1}$.

Step 2. To determine.

We have to find a difference quotient for the given function.

Step 3. Calculation.

We first find$f(x+h)$ by plugging$x+h$ in place of x in $f\left(x\right)$.

$$\begin{gathered} f\left(x\right)=3^{x-1} \\ f(x+h)=3^{(x+h-1)} \end{gathered}$$

Then we find the difference quotient:

$$\begin{gathered} \frac{f(x+h)-f\left(x\right)}{h}=\frac{3^{(x+h-1)}-3^{(x-1)}}{h} \\ =\frac{3^{(x-1)+h}-3^{(x-1)}}{h} \\ =\frac{3^{(x-1)}\cdot 3^h-3^{(x-1)}}{h}[\text{using\hspace{0.17em}\hspace{0.17em}property\hspace{0.17em}\hspace{0.17em}}{a}^{(m+n)}=a^m\cdot a^n] \\ =\frac{3^{(x-1)}\left(3^h-1\right)}{h}\left[\text{factor\hspace{0.17em}\hspace{0.17em}out\hspace{0.17em}\hspace{0.17em}}{3}^{(x-1)}\right] \\ =3^{x-1}\left(\frac{3^h-1}{h}\right) \end{gathered}$$

Hence, it is proved that $\frac{f(x+h)-f\left(x\right)}{h}=3^{x-1}\left(\frac{3^h-1}{h}\right)$.