Question

In Problems $85-88$, determine the exponential function whose graph is given.

Short answer

$85-88$The exponential function is$f\left(x\right)=-6^x$.

Step-by-step solution

Step 1. Given information.

The points through which graph passing are $(-1,-\frac{1}{6}),(0,-1),(1,-6),(2,-36)$.

The function f(x) is an exponential function then for any real numberx, $\frac{f(x+1)}{f\left(x\right)}=a$, $a>0$.

So the exponential function can be written as$f\left(x\right)=a^x$.

Step 2. Substitute x=-1 in f(x+1)f(x).

This gives

$\frac{f(-1+1)}{f(-1)}=\frac{f\left(0\right)}{f(-1)}$

From the graph,

$f\left(0\right)=-1,f(-1)=-\frac{1}{6}$

which gives

$$\begin{gathered} \frac{f\left(0\right)}{f(-1)}=\frac{-1}{-{\displaystyle \frac{1}{6}}} \\ =6 \end{gathered}$$

Step 3. Substitute x=0 in f(x+1)f(x).

This gives

$\frac{f(0+1)}{f\left(0\right)}=\frac{f\left(1\right)}{f\left(0\right)}$

From the graph,

$f\left(0\right)=-1,f\left(1\right)=-6$

which gives

$$\begin{gathered} \frac{f\left(1\right)}{f\left(0\right)}=\frac{-6}{-1} \\ =6 \end{gathered}$$

Step 4. Finding exponential function.

Similarly, $\frac{f\left(2\right)}{f\left(1\right)}=6$.

So, the ratio of the consecutive values is a constant $6$, which is the base a of the exponential function. So, the exponential function may be $6^x$.

But, it can be seen that given graph is lying below x-axis i.e. for all x, y is always negative.

So, the exponential function of the given graph is $f\left(x\right)=-6^x$.