Real Function/Examples/Arbitrary Function 1/Mistake

Source Work

1963: Morris Tenenbaum and Harry Pollard: Ordinary Differential Equations:

Chapter $1$: Basic Concepts:

Lesson $2 \text B$: The Meaning of the Term Function of One Independent Variable

Mistake

The relationship between two variables $x$ and $y$ is the following. If $x$ is between $0$ and $1$, $y$ is to equal $2$. If $x$ is between $2$ and $3$, $y$ is equal to $\sqrt x$. The equations which express the relationship between the two variables are, with the end points of the interval included,

\(\text {(a)}: \quad\)

\(\ds y\)

\(=\)

\(\ds 2,\)

\(\ds 0 \le x \le 1,\)

\(\ds y\)

\(=\)

\(\ds \sqrt x,\)

\(\ds 2 \le x \le 3.\)

These two equations now define $y$ as a function of $x$. For each value of $x$ in the specified intervals, a value of $y$ is determined uniquely. The graph of this function is shown in Fig. $2.211$. Note that these equations do not define $y$ as a function of $x$ for values of $x$ outside the two stated intervals.

Figure $2.211$

Correction

The shape of the second part of the graph is incorrect.

It has been depicted as a convex function, in shape more like a square function than a square root.

The square root is a concave function.

The actual shape of the function in question can be found in the depiction of this arbitrary function.

Sources

• 1963: Morris Tenenbaum and Harry Pollard: Ordinary Differential Equations ... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $2 \text B$: The Meaning of the Term Function of One Independent Variable: Example $2.21$