AHL 2.14—Odd and even functions, self-inverse, inverse and domain restriction Questionbank

State the range of $g$.

Find the coordinates of the point where the graph of $y=g(x)$ intersects the $y$-axis.

Show that $f(x)=x+p+3+\dfrac{q+3p+9}{x-3}$ and state the equation of the oblique asymptote.

Prove that $f$ is neither even nor odd for any $p,q$.

Find the conditions on $p$ and $q$ such that the graph of $f$ has exactly one $x$-intercept, and give the coordinate of this intercept in terms of $p$.

For $q+3p+9\ne0$ and $\Delta>0$, solve the inequality $f(x)\ge x+p+3$.

Sketch the graph of $y=f(x)$, indicating all asymptotes with their equations, the y-intercept, and any local maximum or minimum points.

Show that $g^{-1}(x)=3+\sqrt{\frac{4 x+1}{x}}$.

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

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

Solve $(h \circ g)(a)=\frac{\pi}{6}$, giving the answer in the form $p+q\sqrt{r}$, where $p, q, r \in \mathbb{Z}^{+}$.

Hence, or otherwise, find the x-coordinates of the points where the graphs of $y=f(x)$ and $y=g^{-1}(x)$ intersect for $x>5$, and determine the intervals where $f(x)>g^{-1}(x)$.

Divide to obtain $f(x)=Q(x)+\dfrac{R}{x+3}$ and state the equation of the slant asymptote.

Show that $f$ is neither even nor odd for any $p,q$.

Find the conditions on $p$ and $q$ for the graph of $f$ to have exactly one $x$-intercept, and state the coordinates of the intercept in each case.

For $q-3p+9\ne0$ and $\Delta>0$, solve the inequality $f(x)\ge x+p-3$.

Find $f^{-1}(x)$ and its domain.

Determine when $f$ is self-inverse.

For $a=0$, find fixed points and solve $f(x)\ge x$.

For $a=1$, solve $f(x)=\sqrt{x+5}$ numerically to 3 s.f., stating the number of solutions.

Compute $f^{-1}(x)$ and its domain.

For which $k$ is $f$ self-inverse?

Let $g_k(x)=\dfrac{kx+1}{x-k}$. Prove self-inverse and find fixed points.

For $k>0$, solve $g_k(x)\ge x$. For $k=2$, solve $g_k(x)=\ln(1+x)$ to 3 s.f., noting the number of roots.

Find $f^{-1}(x)$.

Show that $f$ is a self-inverse function if and only if $m-q=0$.

Let $g_s(x)=\dfrac{s x+1}{x-s}$ where $s\ne0$. Prove that $g_s$ is a self-inverse function and find its fixed points.

For $s>0$, solve the inequality $g_s(x)\ge x$.

State the range of $g$.

Describe the transformation that maps $y=\ln x$ to $y=g(x)$.

Find the coordinates of the point where the graph of $y=g(x)$ intersects the line $y=\ln 2$.

Find the coordinates of the point of inflection on the graph of $y=f(x)$.

Draw a set of axes showing $x$ and $y$ values between $-2$ and $4$. On these axes, sketch the graph of $y=f(x)$, showing clearly any axis intercepts and giving the equations of any asymptotes.

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

Express $f(x)$ in the form $x+a+\frac{b}{x+6}$, where $a, b \in \mathbb{R}$, and hence find the equation of the oblique asymptote.

Show that $f$ is neither even nor odd.

Find the values of $p$ and $q$ such that the graph of $f$ has exactly one $x$-intercept, and state the coordinates of this intercept.

Given that $q-6p+36 \ne 0$ and $\Delta > 0$, solve the inequality $f(x) \ge x+p-6$.