Question

NO LINKS!! Please help me with this problem. Part 11ff​

Answer 1

Explanation:

1. if x=-4, then the value of the given function is f(x)=7/0 - impossible to divide by zero; in this case the correct answer is 'any number except -4';

2. if to see the behavior of the given function near x=-4, then:

`\lim_(x \to -4^+)( (7)/(x+4) )=+ \infty;\\ \lim_(x \to -4^-)((7)/(x+4))=- \infty.`

3. correct answers are marked with green colour (see the attachment).

Answer 2

Domain: all real numbers x except x = -4

`\textsf{End behaviour}: \quad f(x) \rightarrow - \infty \; \textsf{as} \; x \rightarrow -4^(-), \;f(x) \rightarrow \infty \; \textsf{as} \; x \rightarrow -4^(+)`

Explanation:

Given function:

`f(x)=(7)/(x+4)`

The domain of a function is the set of all possible input values (x-values).

When the denominator of a rational function is zero, the function is undefined.

To find the value(s) of x for which the function is undefined, set the denominator to zero and solve for x:

`\implies x+4=0`

`\implies x+4-4=-4`

`\implies x=-4`

Therefore, the domain of the given function is:

• all real numbers x except x = -4

The vertical asymptote(s) of a rational function are the values of x for which the denominator is zero.

Therefore, there is a vertical asymptote at x = -4.

As x gets very close to x = -4 from the negative side, the function approaches -∞ because the denominator will be an extremely small negative number.

As x gets very close to x = -4 from the positive side, the function approaches ∞ because the denominator will be an extremely small positive number.

Therefore, the behaviour of the function near the excluded x-value is:

• 
  `f(x) \rightarrow - \infty \; \textsf{as} \; x \rightarrow -4^(-), \;f(x) \rightarrow \infty \; \textsf{as} \; x \rightarrow -4^(+)`