Suppose f(x)=(x^2-6 x-16)/((x^2-7 x-8) √(x^2-4)) (a) For what numbers x is f defined? (b) For wh

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Suppose f(x)=(x^2-6 x-16)/((x^2-7 x-8) √(x^2-4)) (a) For...

Suppose $f(x)=\frac{x^{2}-6 x-16}{\left(x^{2}-7 x-8\right) \sqrt{x^{2}-4}}$
(a) For what numbers $x$ is $f$ defined?
(b) For what numbers $x$ is $f$ discontinuous?
(c) Which discontinuities found in (b) are removable?

Key Concepts

Removable Discontinuities

A removable discontinuity is a gap in the graph of a function that can potentially be 'filled in' by appropriately defining the function at that point. It occurs when the limit exists at the point of discontinuity, implying that the discontinuity does not affect the overall tendency of the function near that value.

Discontinuities of Functions

Discontinuities occur at points where a function is not continuous. They arise when the function is undefined, typically due to division by zero, and are identified by analyzing the points at which the function’s expression fails to have a definite limit or value.

Restrictions Imposed by Square Roots

Expressions under a square root must be non-negative to output real numbers. Additionally, if the square root appears in a denominator, it leads to further restrictions because its value must be strictly positive to avoid division by zero.

Domain of a Function

The domain refers to the set of all input values for which a function is defined. When determining the domain, one must account for restrictions imposed by the function’s components, such as ensuring denominators are not zero and radicands of even-order roots are non-negative.

Rational Functions and Division by Zero

In rational functions, any value that renders the denominator zero must be excluded from the domain. This involves solving for the zeros of the denominator and ensuring these values are not included in the function’s valid inputs.

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