IB Mathematics Analysis and Approaches Topic AHL…

Let $f(x)=\left|\frac{2 x+1}{x-3}\right|,\ x \in \mathbb{R} \,|\,\{3\}$ , and $g(x)=\arctan (|x-1|),\ x \in \mathbb{R}$ . The inverse of $f$ restricted to $x>3$ is $f^{-1}$ .

Find the equations of the vertical and horizontal asymptotes of $y=f(x)$ .

On the axes below, the graph of $y=f(x)$ for . the asymptotes, intercepts, and any local minima or maxima. ![](A set of coordinate axes suitable for sketching on , with -values shown up to at least , and with the vertical line visible.)

Let $a$ be a positive real number.

Solve the inequality $|x-3| < 2x$ .

The function $f$ is defined by $f(x) = x^2 - 4x$ for .

The figure displays the graph of $y=f(x)$ , with $f(x)=x+3$ . It cuts the -axis at and the -axis at .

Solve the inequality $|2x - 1| > x + 3$ for .

Show that $f^{-1}(x)=\frac{3 x+1}{x-2}$ for $x>2$ .

Solve the inequality $f(x) \leq 1$ . Give your answer in interval notation.

Show that the solution of $|x-a| < 2x$ is $x > \frac{a}{3}$ .

why $|x-a| < 2x$ has no negative solutions, and verify the general result of part (b) against your answer to part (a).

Find the $x$ -intercepts and the coordinates of the vertex of $y = f(x)$ .

On the same set of axes, the graphs of $y = f(x)$ and $y = [f(x)]^2$ , indicating clearly the $x$ -intercepts and the $y$ -coordinate of the local maximum of $y = [f(x)]^2$ .

the coordinates of the local minima of $y = [f(x)]^2$ and why these points coincide with the $x$ -intercepts of $f$ .

Solve the equation $[f(x)]^2 = 4f(x)$ algebraically.

On a separate set of axes, the graph of $y = f(|x|)$ , indicating all $x$ -intercepts, the $y$ -intercept and any turning points.

On the same coordinate axes, the graph of $y=|f(x)|$ , labelling the coordinates of any intercepts and vertices.

Solve the inequality $|f(x)| \leq 3$ , giving your answer in interval notation.

IB Mathematics Analysis and Approaches AHL 2.16—graphing Modulus Equations and Inequalities Questionbank

IB Mathematics Analysis and Approaches AHL 2.16—graphing Modulus Equations and Inequalities Questionbank

IB Mathematics Analysis and Approaches AHL 2.16—graphing Modulus Equations and Inequalities Questionbank

IB Mathematics Analysis and Approaches AHL 2.16—graphing Modulus Equations and Inequalities Questionbank

IB Mathematics Analysis and Approaches (AA) Topic AHL 2.16—graphing Modulus Equations and Inequalities Questionbank