Comprehensive assessment covering inequalities, inverse and composite functions, parametric curves, and graphing techniques such as $y = |f(x)|$, $y = f(|x|)$, $y = \frac{1}{f(x)}$ and $y = \sqrt{f(x)}$.
The inverse function of $f(x) = 2x + 3$ is:
The solution to $|x - 2| < 5$ is:
The solution to $\frac{x}{x - 1} > 0$ is:
For a function to have an inverse that is also a function, it must be:
The graph of $y = f^{-1}(x)$ is the reflection of $y = f(x)$ in:
If $f(x) = x^2$ is restricted to $x \geq 0$, then $f^{-1}(x)$ equals:
If $f(x) = x + 1$ and $g(x) = x^2$, then $(f \circ g)(x)$ equals:
The solution to $x^2 - x - 6 \leq 0$ is:
Which domain restriction allows $f(x) = x^2 - 4$ to have an inverse function?
A self-inverse function satisfies $f(f(x)) = x$. Which function is self-inverse?
A curve has parametric equations $x = t^2$, $y = 2t$. Its Cartesian equation is:
The parametric equations $x = \cos\theta$, $y = \sin\theta$ describe:
The graph of $y = |f(x)|$ is obtained from $y = f(x)$ by:
The graph of $y = f(|x|)$ is always:
The graph of $y = \frac{1}{f(x)}$ has a vertical asymptote wherever:
1. Solve the inequality $\frac{2x + 1}{x - 3} \geq 1$, showing all working. (3 marks)
2. Find the inverse function of $f(x) = \frac{x + 2}{x - 1}$, stating its domain. (3 marks)
3. Sketch $y = |x^2 - 4|$, showing all intercepts and the shape of the graph. (3 marks)
4. Given $f(x) = 3x - 1$ and $g(x) = x^2 + 2$, find both $(f \circ g)(x)$ and $(g \circ f)(x)$. (3 marks)
5. A curve has parametric equations $x = 2t$, $y = t^2 - 1$. Find its Cartesian equation. (2 marks)
6. Solve $|2x - 1| \geq 5$. (2 marks)
7. Show that $f(x) = \frac{3}{x}$ is self-inverse by verifying $f(f(x)) = x$. (2 marks)
8. Restrict the domain of $f(x) = (x - 2)^2$ so that an inverse exists, then find $f^{-1}(x)$. (3 marks)
9. Solve $x^2 - 3x - 10 > 0$ using a sign diagram. (3 marks)
10. The line $y = x - 2$ is the graph of $y = f(x)$. Sketch $y = \frac{1}{f(x)}$, clearly indicating the vertical and horizontal asymptotes. (4 marks)