DP Mathematics: Analysis and Approaches Questionbank

Show that $f$ is an even function.

Show that $f$ is an odd function.

Show that $g^{-1}\left(x\right)=1+\frac{\sqrt{4x^2+x}}{x}$.

State the domain of $g^{-1}$.

Show that $f$ is an odd function.

Show that the inverse function of $f$ is given by $f^{-1}\left(x\right)=\sqrt{x^2+1}$.

Find the largest value of $a$ such that $f$ has an inverse function.

For this value of $a$, find an expression for $f^{-1}\left(x\right)$, stating its domain.

Describe the transformation by which $f\left(x\right)$ is transformed to $g\left(x\right)$.

State the range of $g$.

Sketch the graphs of $y=f\left(x\right)$ and $y=g\left(x\right)$ on the same axes, clearly stating the points of intersection with any axes.

Find the coordinates of P.

Hence, or otherwise, find the coordinates of the point of inflexion on the graph of $y=f\left(x\right)$.

sketch the graph of $y=f\left(x\right)$, showing clearly any axis intercepts and giving the equations of any asymptotes.

sketch the graph of $y=f^{-1}\left(x\right)$, showing clearly any axis intercepts and giving the equations of any asymptotes.

Hence, or otherwise, solve the inequality $f\left(x\right)>f^{-1}\left(x\right)$.

Write down the range of $f$.

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

Determine whether $f_n$ is an odd or even function, justifying your answer.

For $a=-\frac{\pi }{2}$, sketch the graph of $y=g\left(x\right)$. Indicate clearly the maximum and minimum values of the function.

Hence or otherwise, find an expression for the derivative of $f_n(x)$ with respect to $x$.

By using mathematical induction, prove that

$f_n(x)=\frac{\sin 2^{n+1}x}{2^n\sin 2x},x\ne \frac{m\pi }{2}$ where $m\in \mathbb{Z}$.

Show that, for $n>1$, the equation of the tangent to the curve $y=f_n(x)$ at $x=\frac{\pi }{4}$ is $4x-2y-\pi =0$.