Chapter 1, Problem 2RE

Solution

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

The option $\text{f}$ , $f\left(x\right)=\left|x-2\right|$ , seems to be the best match of the function represented in the graph.

Given the graph of a function:

  

Concept Used:

The modulus function:

The modulus function $f\left(x\right)=\left|x\right|$ has the graph given below.

  

Horizontal shift (translation) of a function:

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

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

Determining the function:

Observe the graph of the function, it is the graph of the basic modulus function shifted to the right by $2$ units.

Now, the basic modulus function $f\left(x\right)=\left|x\right|$ is shifted to the right by $2$ units when it is applied with the transformation $f\left(x\right)=\left|x-2\right|$ .

Thus, the function represented by the graph is $f\left(x\right)=\left|x-2\right|$ .

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

Conclusion:

The option $\text{f}$ , $f\left(x\right)=\left|x-2\right|$ , seems to be the best match of the function represented in the graph.