Question 18

Function $f$ is defined as $f(x) = \frac{1}{\sqrt{x^2 - x + n}}$ with $f(-2) = \frac{1}{2}$ .

How many of the following four statements are true based on the information above?

$n$ is an odd number.
$0$ is a member of the range of function $f(x)$ .
The function $f$ has a value of $\frac{1}{4}$ when $x = -4$ .
The domain of function $f$ is $x < -1 \cup x > 2$ .

Explanation

The first step is to find the value of $n$ using the information $f(-2) = \frac{1}{2}$ .

Finding the value of n

Substitute $x = -2$ into the function equation:

f(-2) = \frac{1}{\sqrt{(-2)^2 - (-2) + n}}

\frac{1}{2} = \frac{1}{\sqrt{4 + 2 + n}}

\frac{1}{2} = \frac{1}{\sqrt{6 + n}}

Square both sides:

\left(\frac{1}{2}\right)^2 = \left(\frac{1}{\sqrt{6 + n}}\right)^2

\frac{1}{4} = \frac{1}{6 + n}

4 = 6 + n

n = -2

So, the function $f(x)$ is:

f(x) = \frac{1}{\sqrt{x^2 - x - 2}}

Now let's check the truth of each statement.

Statement 1

$n$ is an odd number. Since $n = -2$ (even), statement 1 is False.

$0$ is a member of the range of function $f(x)$ . The function $f(x) = \frac{1}{\sqrt{\dots}}$ is a fraction with a numerator of 1. This fraction value will never reach zero because the numerator is non-zero. So, $0$ is not in the range. Statement 2 is False.

Statement 3

The function $f$ has a value of $\frac{1}{4}$ when $x = -4$ . Calculate $f(-4)$ :

f(-4) = \frac{1}{\sqrt{(-4)^2 - (-4) - 2}}

= \frac{1}{\sqrt{16 + 4 - 2}}

= \frac{1}{\sqrt{18}}

= \frac{1}{3\sqrt{2}}

Since $\frac{1}{3\sqrt{2}} \neq \frac{1}{4}$ , statement 3 is False.

Statement 4

The domain of function $f$ is $x < -1 \cup x > 2$ . For the function to be defined, the denominator must not be zero and the value inside the square root must be positive. Since the root is in the denominator, the condition is:

x^2 - x - 2 > 0

Factor the quadratic inequality:

(x - 2)(x + 1) > 0

The zeroes are $x = 2$ and $x = -1$ . Testing the number line, we find the positive region is in the interval $x < -1$ or $x > 2$ .

So, the domain of $f$ is $\{x \mid x < -1 \text{ or } x > 2\}$ . Statement 4 is True.

Conclusion

There is only 1 true statement, which is statement 4.

Command Palette

Number 18