Questions covered in Hard

Review the question prompts below before starting the timed test. Try each problem independently, then use the interactive test to record your answers and review the worked solutions.

1. The domain of the function $f(x) = \frac{1}{\sqrt{x+|x|}}$ is
2. The range of the function $f(x) = \frac{x}{|x|}$ is
3. If $f(x) = ax + b$, where $a$ and $b$ are integers, $f(-1) = -5$ and $f(3) = 3$, then $a$ and $b$ are equal to
4. The domain of the function $f(x) = \sqrt{1-x^2}$ is
5. Let $n(A) = m$ and $n(B) = n$. Then the total number of non-empty relations that can be defined from A to B is
6. Find the domain of the function $f(x) = \frac{x^2+2x+1}{x^2-8x+12}$.
7. Find the range of the function $f(x) = 1 - |x-2|$.
8. If $f(x) = x^2 - 3x + 1$, find $x \in R$ such that $f(2x) = f(x)$.
9. Find the domain and range of the real function $f(x) = \frac{1}{\sqrt{9-x^2}}$.
10. If $f(x) = \frac{x+1}{x-1}$, prove that $f(f(x)) = x$.
11. Find the domain and range of the function $f(x) = \frac{x}{1+x^2}$.
12. Case Study: A function $f$ is defined by $f(x) = 1-x$ for $x < 0$, $f(x) = 1$ for $x = 0$, and $f(x) = x+1$ for $x > 0$.
    (i) Draw the graph of $f(x)$.
    (ii) Find the value of $f(-2) + f(2)$.
    (iii) Find the range of the function.