Indefinite Integration
Functions
MJAT None
Grade 12
Question:
Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = \frac{x^2 - 3x - 6}{x^2 + 2x + 4}$. Then which of the following statements is (are) TRUE ?
A) $f$ is decreasing in the interval $(-2, -1)$
B) $f$ is increasing in the interval $(1, 2)$
C) $f$ is onto
D) Range of $f$ is $[-\frac{3}{2}, 2]$
)
Step-by-Step Solution
Key Concept: A function is undefined at a point where its denominator is zero, and it is not continuous at that point.
$$f(x) = \frac{x^2 - 3x - 6}{x^2 + 2x + 4}$$
$$f(x) = \frac{(x - 6)(x + 1)}{(x + 2)(x + 2)}$$
$$f(x) = \frac{x - 6}{x + 2}$$
The given function $f(x)$ is undefined at $x = -2$.
Since the function $f(x)$ is undefined at $x = -2$, it is not continuous at $x = -2$.
Therefore, the correct option is that the function $f(x)$ is undefined at $x = -2$.
Correct Answer: D