f, of, x, equals, start fraction, 2, x, plus, 9, divided by, 2, x, minus, 7, end fractionf(x)= 2x−72x+9​

Understanding the Function

The function you've provided seems to be incorrectly represented. Based on the provided elements, it appears that the correct function format should be $f(x) = \frac{2x + 9}{2x - 7}$.

Key Components

Given:

• $f(x) = \frac{2x + 9}{2x - 7}$

This is a rational function, where:

• The numerator is $2x + 9$.
• The denominator is $2x - 7$.

Domain of the Function

To determine the domain, we must ensure the denominator is not equal to zero:

1. Set the denominator equal to zero: $2x - 7 = 0$
2. Solve for $x$: $2x = 7 \implies x = \frac{7}{2}$ The domain of the function is all real numbers except $x = \frac{7}{2}$.

Vertical Asymptote

The vertical asymptote of the function occurs where the denominator equals zero:

• Therefore, there is a vertical asymptote at $x = \frac{7}{2}$.

Final Answer

The function is correctly represented as: $f(x) = \frac{2x + 9}{2x - 7}$

• Domain: All $x \in \mathbb{R}$ except $x = \frac{7}{2}$.
• Vertical Asymptote: $x = \frac{7}{2}$.