Exercise
Determine the domain of the function:
f(x)=\sqrt{\frac{x^2-x-2}{x^2-4x-21}}
Final Answer
Solution
Let’s find the domain of the function:
f(x)=\sqrt{\frac{x^2-x-2}{x^2-4x-21}}
Because there is a denominator, the denominator must be different from zero:
x^2-4x-21\neq 0
Also, there is a square root, so the expression inside the root must be non-negative:
\frac{x^2-x-2}{x^2-4x-21}\geq 0
Let’s look at the second inequality. When is a fracture greater than or equal to zero? When both the numerator and denominator are positive or when both are negative. Let’s look at the first case – the numerator is positive or zero and the denominator is positive (because denominator cannot be zero):
x^2-x-2\geq 0\text{ and }x^2-4x-21> 0
We got two quadratic inequalities. To solve them, they must be broken down into factors. We use the quadratic formula for both inequalities and get the factorizations:
x^2-x-2=(x+1)(x-2)
x^2-4x-21=(x+3)(x-7)
We solve the first inequality:
x^2-x-2=(x+1)(x-2)\geq 0
Therefore, its roots are -1, 2 (this means that the graph goes through the x-axis at these points) and the parabola (the graph) in the shape of a bowl (“smiling” parabola), because the coefficient of the square expression is positive. According to the inequality sign, one should check when the graph is on or above the x-axis. It happens when
x\leq -1\text{ or }x\geq 2
Now, we solve the second inequality:
x^2-4x-21=(x+3)(x-7)>0
Therefore, its roots are -3, 7 and the parabola in the shape of a bowl (“smiling” parabola), because the coefficient of the square expression is positive. According to the inequality sign, one should check when the graph is on or above the x-axis. It happens when
x< -3\text{ or }x>7
We intersect (“and”) both results, because we want both a positive numerator and a positive denominator) and get the solution for the first case:
x<-3\text{ or } x>7
Now let’s look at the second case: both the numerator and the denominator are negative, that is:
x^2-x-2=(x+1)(x-2)<0
and
x^2-4x-21=(x+3)(x-7)<0
We solve the first inequality. According to the inequality sign, one should check when the graph is below the x-axis. The graph pass through the x-axis at the points: 1-, 2, so we get the solution:
-1<x<2
Now, we solve the second inequality. According to the inequality sign, one should check when the graph is below the x-axis. The graph pass through the x-axis at the points: 1-, 2, so we get the solution:
-3<x<7
We intersect (“and”) both results and get
-1<x<2
And the final answer is the union (“or” relationship) of both solutions – the solution of the first case and the solution of the second case – because they cannot occur together (either the numerator and denominator are both positive or are negative) and we get
x<-3\text{ or } -1<x<2\text{ or }x>7
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