Chapter 1.2, Problem 2E

Solution

To find: whether the formula determines $y$ as a function of $x$ and explain.

  $y=x^2\pm 3$ is not a function.

Given Information:

Equation $y=x^2\pm 3$ is given.

Concept Used:

The function is defined as a special type relation from a set $A$ to $B$ such that for every element $x$ of the set $A$ , there exist only one image $y$ in set $B$

Written as, $f\left(x\right)=y$

Calculation:

Observe that when any value of number $x$ is substituted into the equation, two values of $y$ is obtained

Thus, from the definition of function mentioned there must exist one and only image which is not the case here.

So, the function is an ambiguous formula like $f\left(x\right)=x^2\pm 3$ and not a function as each $y$ has two values for each positive value of $x$ .

Example:

Put $x=1$ in the given equation,

  $\begin{array}{l}f(x)=y\\ y=x^2\pm 3\\ y=\left(1^2+3\right)\text{or}\left(1^2-3\right)\\ y=4\text{or}-2\end{array}$

Therefore, $y=x^2\pm 3$ is not a function.