IB Mathematics Analysis and Approaches Topic SL…

A function $f$ is defined by $f(x)=\cos ^{-1}\left(\frac{1}{x^{2}+1}\right)$ for $x\in\mathbb{R}$ . A function $g$ is defined by $g(x)=\cos ^{-1}\left(\frac{1}{x^{2}+1}\right)$ for $x\geq 0$ .

State the domain $D$ of $f$ .

Sketch the graph of $y=f(x)$ , showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.

Let $f(x)=e^{kx},\quad g(x)=\ln(x+2),\quad x>-2,\ k>0.$ It is given that .

Let $p\in\mathbb{R}$ and $q_p(x)=x^2-(p+8)x+(2p+8)$ .

For a real parameter $p$ , consider the quadratic $q_p(x)=x^2-(p+2)x+(2p-5).$

Consider the function $f(x)=\frac{1}{x+3}$ .

Explain why $f$ is an even function.

State why the inverse function $f^{-1}$ does not exist.

Let $g$ be the restriction of $f$ to $x\geq 0$ . Find the inverse function $g^{-1}(x)$ and state its domain.

Determine $k$ correct to 3 significant figures.

Find the composites and their maximal domains.

Solve $(x+2)^k=\ln(2+e^{kx})$ for $x\ge0$ (3 d.p.) and justify uniqueness.

Find $(f\circ g)^{-1}(y)$ with domain and range.

Find all real values of $p$ for which $q_p$ has two distinct real roots and both roots are greater than $1$ .

Let $p=3$ (not necessarily satisfying the condition in the previous part) so that $q(x)=x^2-11x+14$ , and . Find the exact roots of .

Solve $q(x)=g(x)$ for $x\ge0$ , giving solutions correct to $3$ d.p., and justify the number of solutions.

Find all real values of $p$ for which $q_p$ has two distinct real roots and both roots are greater than $0$ .

Let $p=4$ so that $q(x)=x^2-6x+3$ , and let (domain ).

Solve $q(x)=g(x)$ for $x\ge0$ to three decimal places. Justify the number of solutions on $x\ge0$ .

Find the domain for the function $f(x)$ .

Find the range for the function $f(x)$ .

Sketch the graph of the function $f(x)$ .

IB Mathematics Analysis and Approaches SL 2.2—functions, Notation Domain, Range and Inverse as Reflection Questionbank

IB Mathematics Analysis and Approaches (AA) Topic SL 2.2—functions, Notation Domain, Range and Inverse as Reflection Questionbank