Question

use long division to rewrite the equation ( g(x) = \frac{6x + 19}{x + 3} ) in the form quotient, plus remainder divided by divisor. then use this form of the function’s equation and transformations of ( f(x) = \frac{1}{x} ) to graph.

write ( g(x) ) in the form quotient, plus remainder divided by divisor.
( g(x) = square )

choose the correct graph below.

a. graph a

b. graph b

c. graph c

d. graph d

Explanation:

Step1: Perform Long Division

Divide \(6x + 19\) by \(x + 3\).
\(6x\div x = 6\) (quotient term). Multiply \(x + 3\) by \(6\): \(6(x + 3)=6x + 18\).
Subtract from \(6x + 19\): \((6x + 19)-(6x + 18)=1\) (remainder).
Thus, \(g(x)=\frac{6x + 19}{x + 3}=6+\frac{1}{x + 3}\).

Step2: Analyze the Graph

The function \(g(x)=6+\frac{1}{x + 3}\) is a transformation of \(f(x)=\frac{1}{x}\):

• Horizontal shift: \(f(x + 3)\) shifts \(f(x)\) left by 3 units.
• Vertical shift: \(6 + f(x + 3)\) shifts up by 6 units.

The vertical asymptote is \(x=-3\) (from \(x + 3 = 0\)), and horizontal asymptote is \(y = 6\) (from the vertical shift).

Check the graphs:

• Asymptotes \(x=-3\) and \(y = 6\) match graph C (vertical asymptote at \(x=-3\), horizontal at \(y = 6\)).

Answer:

\(g(x)=6+\frac{1}{x + 3}\), and the correct graph is C.