Find a value of the constant k, if possible, that will make...

Find a value of the constant $k,$ if possible, that will make the function continuous everywhere. $$ \text { (a) } f(x)=\left\{\begin{array}{ll}{9-x^{2},} & {x \geq-3} \\ {k / x^{2},} & {x<-3}\end{array}\right. $$ $$ \text { (b) } f(x)=\left\{\begin{array}{ll}{9-x^{2},} & {x \geq 0} \\ {k / x^{2},} & {x<0}\end{array}\right. $$

Find a value of the constant $k,$ if possible, that will make the function continuous everywhere.
$$
\text { (a) } f(x)=\left\{\begin{array}{ll}{9-x^{2},} & {x \geq-3} \\ {k / x^{2},} & {x<-3}\end{array}\right.
$$
$$
\text { (b) } f(x)=\left\{\begin{array}{ll}{9-x^{2},} & {x \geq 0} \\ {k / x^{2},} & {x<0}\end{array}\right.
$$

Key Concepts

Parameter Determination for Continuity

When a piecewise-defined function contains an unknown parameter, determining its value typically involves equating the limit from one side of the boundary to the function’s value (or the limit from the other side) at that point. This process ensures that the function does not have a discontinuity at the boundary where the definition changes.

One-Sided Limits

One-sided limits examine the behavior of a function as it approaches a particular point from one side (either the left or the right). They are crucial in determining the continuity of a piecewise function at the boundary points, by verifying that the function approaches the same value from both sides.

Continuity

Continuity of a function at a point means that the function's value and its limit as the input approaches that point are the same, ensuring there is no interruption or jump in the function’s behavior at that point. For a function to be continuous everywhere, it must be continuous at every point in its domain.

Piecewise-Defined Functions

Piecewise-defined functions are functions that are defined by different expressions over different intervals of the domain. Each piece can have its own functional rule, and special attention is needed at the boundary points where the definition changes to ensure properties like continuity are maintained.

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