Question 10

Explanation

Given function $f(x) = \frac{1}{\sqrt{-x^2 + 4x + 21}}$.

Quantity $P = 7$.

To determine quantity $Q$, we need to find the values of $x$ such that $f(x)$ is defined as a real number.

For $f(x)$ to be defined, the denominator must not be zero and the expression under the square root must be greater than zero

Multiply both sides by $-1$ (remember the inequality sign reverses)

Factor the quadratic expression

The roots of $(x - 7)(x + 3) = 0$ are $x = 7$ and $x = -3$.

Using a number line, the inequality $(x - 7)(x + 3) < 0$ is satisfied for $-3 < x < 7$.

So the domain of function $f$ is $-3 < x < 7$.

The integer values of $x$ that satisfy this are the integers in the interval $(-3, 7)$, which are $x = -2, -1, 0, 1, 2, 3, 4, 5, 6$.

The number of such values of $x$ is $9$, so $Q = 9$.

Since $P = 7$ and $Q = 9$, we have $P < Q$.

Therefore, quantity $P$ is less than quantity $Q$.