Question

Which statement describes the behavior of the function f(x)=3x/4-x?

Answer 1

Answer:

The  graph approaches –3 as x approaches infinity. Option a is correct.

Step-by-step explanation:

The given function is

$f(x)=\frac{3x}{4-x}$

We have to find value of function as x approaches infinity. Take limit both sides as x approaches to infinity.

$\lim_{x\rightarrow \infty}f(x)=\lim_{x\rightarrow \infty}\frac{3x}{4-x}$

Taking x common from the denominator.

$\lim_{x\rightarrow \infty}f(x)=\lim_{x\rightarrow \infty}\frac{3x}{x(\frac{4}{x}-1)}$

Cancel out common factor x.

$\lim_{x\rightarrow \infty}f(x)=\lim_{x\rightarrow \infty}\frac{3}{\frac{4}{x}-1}$

Apply limits.

$\lim_{x\rightarrow \infty}f(x)=\frac{3}{\frac{4}{\infty}-1}$

$\lim_{x\rightarrow \infty}f(x)=\frac{3}{0-1}$

$\lim_{x\rightarrow \infty}f(x)=-3$

Therefore the  graph approaches –3 as x approaches infinity.

Answer 2

A. The graph approaches –3 as x approaches infinity.

$f(x) = \frac{3x}{4-x} +3 -3 = \frac{3x+12-3x}{4-x} -3 = \frac{12}{4-x} -3$