Find the Holes in the Graph f(x)=(5x+3)/(x+6)
This problem asks you to determine the locations of any holes in the graph of the given rational function f(x) = (5x+3)/(x+6). A hole in the graph of a rational function occurs at a value of x that causes both the numerator and the denominator to equal zero when the common factor is not actually part of the function's simplified form. To find the holes, you would typically factor both the numerator and denominator of the function (if possible) and look for common factors. If a common factor is identified, its corresponding x-value is where a hole exists, since the function is not defined at that x-value even though it cancels out in the simplified form of the function.
$f \left(\right. x \left.\right) = \frac{5 x + 3}{x + 6}$
Answer
Solution:
Step 1: Identifying Holes in the Graph
A hole in the graph of a rational function occurs where the function is not defined, which is typically due to a common factor in the numerator and denominator that can be canceled out. For the function \( f(x) = \frac{5x + 3}{x + 6} \), there are no common factors between the numerator and the denominator. Therefore, the graph of \( f(x) \) does not have any holes.
Step 2: Conclusion
Since there are no common factors to cancel, the function \( f(x) = \frac{5x + 3}{x + 6} \) is defined for all values of \( x \) except where the denominator is zero, which is at \( x = -6 \). However, this results in a vertical asymptote, not a hole. Thus, we conclude that there are no holes in the graph of the given function.
Knowledge Notes:
Rational Functions: A rational function is a function of the form \( f(x) = \frac{p(x)}{q(x)} \), where \( p(x) \) and \( q(x) \) are polynomials and \( q(x) \neq 0 \). The domain of a rational function includes all real numbers except those that make the denominator equal to zero.
Holes in the Graph: A hole in the graph of a rational function occurs at a point where both the numerator and the denominator have a common factor, which can be canceled. The hole is at the x-value that would make the common factor equal to zero.
Vertical Asymptotes: A vertical asymptote occurs in the graph of a rational function at a value of \( x \) that makes the denominator zero, provided that the numerator does not have a common factor with the denominator at that \( x \)-value. This is where the function approaches infinity or negative infinity.
Simplifying Rational Functions: To identify holes, one must factor the numerator and the denominator of the rational function and cancel out any common factors. If no common factors exist, then there are no holes.
Graph Analysis: When analyzing the graph of a rational function, it is important to identify holes, vertical asymptotes, horizontal asymptotes, and the behavior of the function as \( x \) approaches the asymptotes or holes.
In the given problem, since the numerator \( 5x + 3 \) and the denominator \( x + 6 \) have no common factors, there are no holes in the graph of \( f(x) \). The only point of discontinuity is \( x = -6 \), which is a vertical asymptote, not a hole.
