Chapter 1, Problem 64CR

To determine

To identify: The parent function and the transformation shown in the graph and write an equation for the function represented by the graph.

Answer to Problem 64CR

The parent function of function is $f\left(x\right)=\frac{1}{x}$ and equation for the function represented by the given graph is $g\left(x\right)=\frac{1}{\left(x+4\right)}+4$ .

Explanation of Solution

Given information:

The given function is shown in Figure-1.

Concept used:

  

A parent function is the simplest function of a family of functions represents the definition (or shape) of the family of functions. For example, the family of quadratic functions has the general form $g\left(x\right)=a{x}^2+bx+c$ . The simplest function or parent function of this family is $f\left(x\right)=x^2$ .

Identification:

The shape of given graph (Figure-1) appears similar to the graph of a cubic function $f\left(x\right)=\frac{1}{x}$ whose graph is shown in Figure-2.

If we shift the graph of function $f\left(x\right)=\frac{1}{x}$ to the right along - x axis by 4 unit, and then shifting graph 4 units along + y -axis, we get the graph of given function shown in Figure-1.

The equation of function obtained by shifting the graph of function $f\left(x\right)$ to the left along - x axis by 4 units is given by $f\left(x+2\right)$ , and then shifting graph 4 units along + y -axis,

Thus, equation the transformed function say $g\left(x\right)$ is $g\left(x\right)=f\left(x+4\right)+4$ or $g\left(x\right)=\frac{1}{\left(x+4\right)}+4$

  

The graphs functions $f\left(x\right)=\frac{1}{x}$ and $g\left(x\right)=\frac{1}{\left(x+4\right)}+4$ are shown in Figure-2.

Thus, the parent function of the function whose graph is shown in given figure (Figure-1) is $f\left(x\right)=\frac{1}{x}$ and equation for the function represented by the given graph is $g\left(x\right)=\frac{1}{\left(x+4\right)}+4$ .

Conclusion:

The parent function of function is $f\left(x\right)=\frac{1}{x}$ and equation for the function represented by the given graph is $g\left(x\right)=\frac{1}{\left(x+4\right)}+4$ .