Consider the rational function, 2x13x15 What is(ure) the non-permissible value(s) for this rational function? Describe the behaviour of the graph of this function as x approaches each side of each of the non-permissible value(s).
The Correct Answer and Explanation is :
The rational function you are referring to is ( \frac{2x}{13x + 15} ). To identify the non-permissible values, we need to determine when the denominator equals zero because division by zero is undefined.
Step 1: Set the denominator equal to zero:
[
13x + 15 = 0
]
Now, solve for ( x ):
[
13x = -15
]
[
x = \frac{-15}{13}
]
Non-Permissible Value:
The non-permissible value for this function is ( x = \frac{-15}{13} ).
Step 2: Analyze the behavior of the function as ( x ) approaches ( \frac{-15}{13} ) from each side:
- As ( x \to \frac{-15}{13}^- ) (from the left), the denominator ( 13x + 15 ) becomes a small negative number (since ( x ) is slightly less than ( \frac{-15}{13} ), making ( 13x + 15 ) slightly negative), and the numerator ( 2x ) is negative as well. Thus, the function will approach negative infinity.
- As ( x \to \frac{-15}{13}^+ ) (from the right), the denominator ( 13x + 15 ) becomes a small positive number (since ( x ) is slightly greater than ( \frac{-15}{13} ), making ( 13x + 15 ) slightly positive), and the numerator ( 2x ) is still negative. Therefore, the function will approach positive infinity.
Step 3: Summary of Graph Behavior:
At ( x = \frac{-15}{13} ), the function has a vertical asymptote, meaning that the graph approaches infinity on one side and negative infinity on the other side. This is a classic behavior for rational functions when the denominator is zero.
Thus, the graph of the function will show a vertical asymptote at ( x = \frac{-15}{13} ), and as ( x ) approaches this point from the left, the function heads toward negative infinity, while as ( x ) approaches from the right, it heads toward positive infinity.