Define g(4) in a way that extends g(x) = (x² − 16)/(x² − 3x − 4) to be continuous at x = 4.

Verified step by step guidance

1. First, identify the points where the function g(x) = (x² − 16)/(x² − 3x − 4) is undefined. This occurs when the denominator is zero. Set x² − 3x − 4 = 0 and solve for x.
2. Factor the quadratic equation x² − 3x − 4 = 0. This can be factored as (x - 4)(x + 1) = 0, giving the solutions x = 4 and x = -1. These are the points where the function is undefined.
3. To make g(x) continuous at x = 4, we need to remove the discontinuity by simplifying the expression. Notice that the numerator x² − 16 can be factored as (x - 4)(x + 4).
4. Simplify the expression by canceling the common factor (x - 4) from the numerator and the denominator. This gives us a new function h(x) = (x + 4)/(x + 1) for x ≠ 4.
5. Finally, define g(4) as the limit of h(x) as x approaches 4. Calculate this limit by substituting x = 4 into h(x), resulting in g(4) = (4 + 4)/(4 + 1).

Key Concepts

Continuity of Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. For a function to be continuous at x = 4, we need to ensure that the limit of g(x) as x approaches 4 exists and is equal to g(4). This is crucial for defining g(4) in a way that maintains continuity.

Limits

The limit of a function describes the behavior of the function as it approaches a specific input value. In this case, we need to calculate the limit of g(x) as x approaches 4. If this limit exists, it can be used to define g(4) such that the function remains continuous at that point.

Rational Functions

Rational functions are ratios of polynomials, and they can have points of discontinuity where the denominator equals zero. In the given function g(x) = (x² − 16)/(x² − 3x − 4), we need to analyze the denominator to identify any discontinuities at x = 4 and determine how to redefine g(4) to eliminate this discontinuity.