To identify the horizontal asymptote of the function f(x) = (7x + 1) / (2x – 9), we need to analyze the behavior of the function as x approaches infinity (both positive and negative infinity).
1. **Understanding Horizontal Asymptotes**: Horizontal asymptotes describe the behavior of a function as the input values grow very large or very small. Specifically, they indicate what value the function approaches when x tends towards infinity or negative infinity.
2. **Leading Terms**: In rational functions, the horizontal asymptote is determined by the leading terms of the numerator and the denominator. For our function:
- The numerator’s leading term is 7x.
- The denominator’s leading term is 2x.
3. **Finding the Asymptote**: We compare the degrees of the leading terms. Both the numerator and the denominator are of degree 1 (since both have x to the first power). For rational functions where the degrees of the numerator and denominator are equal, the horizontal asymptote can be found by taking the ratio of the leading coefficients:
Horizontal Asymptote (H.A.) = leading coefficient of numerator / leading coefficient of denominator = 7 / 2
4. **Conclusion**: Thus, the horizontal asymptote of the function is y = 7/2. This means that as x approaches infinity (either positively or negatively), the function f(x) approaches the line y = 3.5.
In summary, the horizontal asymptote for f(x) = (7x + 1) / (2x – 9) is y = 3.5.