Question

Is the graph of the following function a polynomial, exponential, or logarithmic function? Determine the function whose graph is given.

Short answer

• The given graph is a graph of an exponential function
• The function is$f\left(x\right)=2·3^x$

Step-by-step solution

Step 1. Given Information :

Given Figure of Graph:

We want to know

• Is the graph of the following function a polynomial, exponential, or logarithmic function.
• Determine the function whose graph is given.

Step 2. Solution: 

From the graph we can read the following:

• function domain:$f(-\infty ,\infty )$
• range of$f(0,\infty )$
• the function is strictly increasing.

From the above we conclude that the given graph is a graph of an exponential function (which could easily be deduced from the appearance of the graph)

In general, we can look for a record of an exponential function in the form:

$f\left(x\right)=b·a^x+c$

Since we see from the graph that these is no displacement along the y-axis then c = 0, because the graph does not intersect the x-axis in 1 then we know that $b\ne 1$

$f\left(x\right)=b·a^x---\left(1\right)$

We read the points from the graph:

$$\begin{gathered} A(-1,\frac{2}{3}) \\ B(0,2) \\ C(1,6) \end{gathered}$$

By including points B( 0 , 2 ) and C( 1 , 6 ) in (1) we obtain:

$$\begin{gathered} 2=b·a^{0}and6=b·a^1 \\ 2=band6=b·a \\ b=2and6=2·a \\ b=2anda=3 \end{gathered}$$

Conclusion:$f\left(x\right)=2·3^x$