Q. \( g(x) = \dfrac{x^2 – 4x – 21}{x + 13} \).

Problem

Given the function

\[ g(x)=\dfrac{x^{2}-4x-21}{x+13}, \]

we will analyze and simplify it step by step, explaining each operation in detail.

1. Step 1 — Factor the numerator

   Look for two numbers whose product is −21 and whose sum is −4. Those numbers are −7 and 3, because

   \[ (-7)\cdot 3 = -21 \quad\text{and}\quad (-7)+3 = -4. \]

   Therefore the numerator factors as

   \[ x^{2}-4x-21 = (x-7)(x+3). \]

   Substitute this into the definition of g(x):

   \[ g(x)=\dfrac{(x-7)(x+3)}{x+13}. \]

   Observe that none of the numerator factors equals x+13, so there is no factor cancellation.

2. Step 2 — Determine the domain

   The denominator x+13 cannot be zero. Solve x+13 = 0 to find the excluded value:

   \[ x = -13. \]

   Thus the domain is all real numbers except x = −13. In interval notation:

   \[ (-\infty, -13)\cup(-13, \infty). \]

3. Step 3 — Perform polynomial division to find a simplified form and the slant asymptote

   The degree of the numerator (2) is one higher than the degree of the denominator (1), so perform division of x^{2}-4x-21 by x+13. Using synthetic division with root −13:

   Coefficients: 1, −4, −21. Synthetic with −13 yields calculations: bring down 1; multiply 1 by −13 to get −13; add to −4 to get −17; multiply −17 by −13 to get 221; add to −21 to get 200.

   Thus the quotient is x−17 and the remainder is 200. This gives the exact identity

   \[ \dfrac{x^{2}-4x-21}{x+13} = x-17 + \dfrac{200}{x+13}. \]

   Use this form for analysis of end behavior and asymptotes.

4. Step 4 — Vertical asymptote and removable holes

   Because x+13 is zero at x = −13 and there was no factor (x+13) in the numerator to cancel, the function has a vertical asymptote at

   \[ x = -13. \]

   There are no removable holes because no factor cancels.

5. Step 5 — Slant (oblique) asymptote

   From the division result, as x becomes large in magnitude the remainder term 200/(x+13) tends to zero. Therefore the slant asymptote is the quotient part:

   \[ y = x – 17. \]

   You can verify by taking the limit:

   \[ \lim_{x\to\pm\infty}\left(g(x)-(x-17)\right) = \lim_{x\to\pm\infty}\dfrac{200}{x+13} = 0. \]

   The graph approaches the line y = x − 17 for large |x|.

6. Step 6 — x-intercepts and y-intercept

   x-intercepts occur where the numerator is zero (and the denominator is not zero). From the factorization (x-7)(x+3) = 0 we get

   \[ x = 7 \quad\text{and}\quad x = -3. \]

   These are valid because neither equals −13.

   y-intercept is g(0). Compute:

   \[ g(0)=\dfrac{0^{2}-4\cdot 0-21}{0+13}=\dfrac{-21}{13}. \]

7. Step 7 — Behavior summary

   • Domain: all real x except x = −13.
   • Factorized form: \[ g(x)=\dfrac{(x-7)(x+3)}{x+13}. \]
   • Division (useful for graphing): \[ g(x)=x-17+\dfrac{200}{x+13}. \]
   • Vertical asymptote: \[ x=-13. \]
   • Slant asymptote: \[ y=x-17. \]
   • x-intercepts: x = 7 and x = −3.
   • y-intercept: \[ g(0)=-\dfrac{21}{13}. \]
   • No removable holes; the graph never equals its slant asymptote at a finite x because \[ \dfrac{200}{x+13}=0 \] has no finite solution.

This completes the detailed step-by-step analysis of the function g(x) = (x^{2}-4x-21)/(x+13).