Final answer:
To match each rational function with its end behavior, we analyze the effect of x becoming larger for each function: A) heads towards positive infinity, B) approaches zero, C) and D) remain constant at 99, E) approaches 99, and F) approaches 108.
Step-by-step explanation:
The student is asked to match each rational function with its end behavior, specifically as the variable x becomes very large. Let us look at each function individually:
- A) 9x: As x gets larger, the function will also get larger, heading towards positive infinity.
- B) 9/x: As x gets larger, the function approaches zero from the positive side because the numerator is constant while the denominator increases.
- C) 99x/x: Since the x in the numerator and denominator cancels out, this simplifies to 99, a constant function with no change as x increases.
- D) 99 x/x: This appears to be a typo and the intended function might be the same as C), so the behavior would also be a constant 99.
- E) 99x 9/x: This function can be simplified to 99 - 9/x. As x gets larger, the 9/x term approaches zero and the function approaches 99.
- F) 99 9x/x: This appears to be a typo as well, but assuming it means 99 + 9x/x, which simplifies to 99 + 9 as x gets larger, the end behavior is that the function approaches 108.
By understanding how negative exponents flip the base to the denominator and the effect of a constant numerator over a variable denominator, we can deduce the behavior of rational functions as x grows.