Chapter 1, Problem 3RE

Solution

To match the function represented in the graph with its equation.

The option $\text{i}$ , $f\left(x\right)=e^x-1$ , seems to be the best match of the function represented in the graph.

Given the graph of a function:

  

Concept Used:

The exponential function:

The exponential function $f\left(x\right)=e^x$ has the graph given below.

  

Vertical shift (translation) of a function:

Given a function $y=f\left(x\right)$ .

The transformation $y=f\left(x\right)+k$ translates $y=f\left(x\right)$ to the upper side by $k$ units if $k>0$ . On the other hand, the transformation $y=f\left(x\right)+k$ translates $y=f\left(x\right)$ to the lower side by $k$ units if $k<0$ .

Determining the function:

Observe the graph of the function, it is the graph of the basic exponential function shifted to the lower side by $1$ unit.

Now, the basic exponential function $f\left(x\right)=e^x$ is shifted to the lower side by $1$ unit when it is applied with the transformation $f\left(x\right)=e^x-1$ .

Thus, the function represented by the graph is $f\left(x\right)=e^x-1$ .

That is, the option $\text{i}$ represents the graph given in the problem.

Conclusion:

The option $\text{i}$ , $f\left(x\right)=e^x-1$ , seems to be the best match of the function represented in the graph.