Question

Find the domain of the following function and express the answer in
interval notation.
Hint: This problem requires you to solve a non-linear inequality
x-6
f(x) =
√x²+3x+2

Answer 1

Answer:

$(-\infty, -2)\cup(-1,\infty)$

Step-by-step explanation:

Given function:

$f(x)=\dfrac{x-6}{\sqrt{x^2+3x+2}}$

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

A rational function is undefined when its denominator is equal to zero.

Therefore, for function f(x) to be defined, the expression under the square root must be greater than zero:

$x^2+3x+2 > 0$

To solve the inequality, begin by factoring the quadratic:

$x^2+x+2x+2 > 0\\\\x(x+1)+2(x+1) > 0\\\\(x+1)(x+2) > 0$

Since the leading coefficient of x² + 3x + 2 is positive, the quadratic function represents an upward-opening parabola that intersects the x-axis at the points x = -1 and x = -2. Therefore, the interval on which it is positive (above the x-axis) is to the left of x-intercept x = -2 and to the right of x-intercept x = -1.

Therefore, the solution of the inequality is:

$x < -2 \;\;\textsf{or}\;\; x > -1$

So, the domain of the function in interval notation is:

$(-\infty, -2)\cup(-1,\infty)$