Given $$ y=f(x) $$, where $$ f(x)=\frac{2 x+7}{x+3} $$ (a) Divide the numerator by the denominator to write $$ f(x) $$ in the form $$ f(x)= $$ quotient $$ +\frac{ ext { remainder }}{ ext { divisor }} $$. (b) Use transformations of $$ y=\frac{1}{x} $$ to graph the function. Part: $$ \mathbf{0} / 2 $$ Part 1 of 2 (a) Divide the numerator by the denominator to write $$ f(x) $$ in the form $$ f(x)= $$ quotient $$ +\frac{ ext { remainder }}{ ext { divisor }} $$. $$ f(x)=\square $$

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Given $$ y=f(x) $$, where $$ f(x)=\frac{2 x+7}{x+3} $$ (a) Divide the numerator by the denominator to write $$ f(x) $$ in the form $$ f(x)= $$ quotient $$ +\frac{	ext { remainder }}{	ext { divisor }} $$. (b) Use transformations of $$ y=\frac{1}{x} $$ to graph the function. Part: $$ \mathbf{0} / 2 $$ Part 1 of 2 (a) Divide the numerator by the denominator to write $$ f(x) $$ in the form $$ f(x)= $$ quotient $$ +\frac{	ext { remainder }}{	ext { divisor }} $$. $$ f(x)=\square $$

UpStudy AI · Accepted Answer

To divide the numerator by the denominator, we perform polynomial long division for
$$
\frac{2x+7}{x+3}.
$$

1. Divide the leading term $$2x$$ by the leading term $$x$$ to obtain $$2$$.
2. Multiply the divisor by $$2$$: 
   $$
   2(x+3)=2x+6.
   $$
3. Subtract this from the numerator:
   $$
   (2x+7)-(2x+6)=1.
   $$

So, the division yields a quotient of $$2$$ and a remainder of $$1$$. Therefore, we can write
$$
f(x)=2+\frac{1}{x+3}.
$$

The answer is:
$$
\boxed{2+\frac{1}{x+3}}.
$$
$$ f(x) = 2 + \frac{1}{x+3} $$

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