the function f(x) = (6x + 9)/(x - 2) is a rational function. answer parts (a) - (d). a. determine the…

the function f(x) = (6x + 9)/(x - 2) is a rational function. answer parts (a) - (d). a. determine the domain. the domain of f is (-∞,2)∪(2,∞) (type your answer in interval notation. use integers or fractions for any numbers in the expression.) b. find the coordinates of any removable discontinuities (if any exist). select the correct choice and, if necessary, fill in the answer boxes to complete your choice. a. there is a removable discontinuity at (type an ordered - pair, using integers or fractions.) the function simplifies to f(x)= (type your answer in factored form.) b. there are no removable discontinuities. c. check for symmetry. does the graph have y - axis symmetry, symmetry about the origin, or no symmetry? choose the correct answer. a. the graph of f is symmetric about the origin. b. the graph of f is symmetric about the y - axis. c. the graph has no symmetry. d. find the y - intercept or state that the function does not have a y - intercept. select the correct choice and, if necessary, fill in the answer box to complete your choice. a. the y - intercept is y = (simplify your answer. type an integer or a simplified fraction.) b. the function has no y - intercept.

the function f(x) = (6x + 9)/(x - 2) is a rational function. answer parts (a) - (d). a. determine the domain. the domain of f is (-∞,2)∪(2,∞) (type your answer in interval notation. use integers or fractions for any numbers in the expression.) b. find the coordinates of any removable discontinuities (if any exist). select the correct choice and, if necessary, fill in the answer boxes to complete your choice. a. there is a removable discontinuity at (type an ordered - pair, using integers or fractions.) the function simplifies to f(x)= (type your answer in factored form.) b. there are no removable discontinuities. c. check for symmetry. does the graph have y - axis symmetry, symmetry about the origin, or no symmetry? choose the correct answer. a. the graph of f is symmetric about the origin. b. the graph of f is symmetric about the y - axis. c. the graph has no symmetry. d. find the y - intercept or state that the function does not have a y - intercept. select the correct choice and, if necessary, fill in the answer box to complete your choice. a. the y - intercept is y = (simplify your answer. type an integer or a simplified fraction.) b. the function has no y - intercept.

Answer

Explanation:

Step1: Find the domain

The denominator of a rational - function cannot be zero. For $f(x)=\frac{6x + 9}{x - 2}$, set $x-2\neq0$. Solving $x-2\neq0$ gives $x\neq2$. In interval notation, the domain is $(-\infty,2)\cup(2,\infty)$.

Step2: Check for removable discontinuities

A removable discontinuity occurs when a factor in the numerator and denominator cancels out. Since $6x + 9=3(2x + 3)$ and $x-2$ has no common factors with $6x + 9$, there are no removable discontinuities.

Step3: Check for symmetry

  1. Y - axis symmetry: Replace $x$ with $-x$ in $f(x)$. We get $f(-x)=\frac{6(-x)+9}{-x - 2}=\frac{-6x + 9}{-x - 2}$. Since $f(-x)\neq f(x)$, the function is not symmetric about the y - axis.
  2. Origin symmetry: Replace $x$ with $-x$ and check if $f(-x)=-f(x)$. $-f(x)=-\frac{6x + 9}{x - 2}=\frac{-6x - 9}{x - 2}$. Since $f(-x)\neq - f(x)$, the function is not symmetric about the origin. So, the graph has no symmetry.

Step4: Find the y - intercept

The y - intercept is found by setting $x = 0$ in $f(x)$. Substitute $x = 0$ into $f(x)=\frac{6x+9}{x - 2}$, we get $f(0)=\frac{6\times0 + 9}{0 - 2}=-\frac{9}{2}$.

Answer:

a. $(-\infty,2)\cup(2,\infty)$ b. B. There are no removable discontinuities. c. C. The graph has no symmetry. d. A. The y - intercept is $y=-\frac{9}{2}$