IB Mathematics Analysis and Approaches Topic AHL…

Consider the function $f(x)=\ln (2 x-1)-\sin (x)$ , where $x>\frac{1}{2}$ .

(a) the graph of $y=f(x)$ for $0.5<x \leq 4$ , showing the intercepts and the behavior at $x=\frac{1}{2}$ .

(b) Solve the inequality $\ln (2 x-1)<\sin (x)$ .

Let $f(x) = \dfrac{x + 3}{x - 1}$ , for $x \neq 1$ .

(a) Find:

(i) $f(0)$ ;

(ii) the equation of the vertical asymptote of $y = f(x)$ ;

(iii) the equation of the horizontal asymptote of $y = f(x)$ .

(b) Find $f^{-1}(x)$ , stating its domain.

(e) Find all values of $x$ for which $|f(x)| \leq 1$ .

(f) Find all values of $x$ for which $f(|x|) \leq 1$ .

Solve $\sqrt{3x+4} \leq 4-x$ for $x$ .

Solve $\sqrt{2x+5} \leq 5-x$ for $x$ .

Consider the function $f(x)=2-\sin x, \quad 0 \leq x \leq 2\pi$ .

Solve the inequality: $f(x)\geq \frac{3}{2}$ .

(d) the graph of $y=f(x)$ , labelling the intercepts with the axes, the asymptotes, and a line of symmetry.

IB Mathematics Analysis and Approaches AHL 2.15—solutions of Inequalities Questionbank