Chapter 0.5, Problem 5E

To determine

Whether the function of given graph is one to one.

Answer to Problem 5E

The function $y=-3x^3$ is one-to-one.

Explanation of Solution

Given data:

The graph of the function $y=-3x^3$ is shown in Fig. (1).

  

Method used:

Suppose the function $y=f\left(x\right)$ is not a one -to-one function. Then there exists $x_1,x_2,\left(x_1\ne x_2\right)$ such that $y=f\left(x_1\right)$ and $y=f\left(x_2\right)$ , i.e. the points $\left(x_1,y\right)$ and $\left(x_2,y\right)$ lie on it. In other words, a line parallel to x -axis intersect graph of the function at two points.

Calculation:

The graph of function $y=-3x^3$ and a line AB drawn parallel to x -axis is shown in Fig. (2).

  

The line through points ‘A’ parallel to x -axis intersect the graph of the function $y=-3x^3$ at one point $\text{A}\left(x_1,y\right)$ . And another line through points B parallel to x -axis also intersect the graph of the function $y=-3x^3$ at one point $\text{B}\left(x_2,y\right)$ . Therefore, the function $y=-3x^3$ is one-to-one.