DP Mathematics: Analysis and Approaches Questionbank

Write down an expression for $f\text{'}\left(x\right)$.

Hence, given that $f^{-1}$ does not exist, show that $b^2-3ac>0$.

Show that $g^{-1}$ exists.

$g\left(x\right)$ can be written in the form $p(x-2{)}^3+q$ , where $p, q \in \mathrm{ℝ}$.

Find the value of $p$ and the value of $q$.

Hence find $g^{-1}\left(x\right)$.

State each of the transformations in the order in which they are applied.

Sketch the graphs of $y=g\left(x\right)$ and $y=g^{-1}\left(x\right)$ on the same set of axes, indicating the points where each graph crosses the coordinate axes.

State the equation of the vertical asymptote on the graph of $y=f\left(x\right)$.

State the equation of the horizontal asymptote on the graph of $y=f\left(x\right)$.

Use an algebraic method to determine whether $f$ is a self-inverse function.

Sketch the graph of $y=f\left(x\right)$, stating clearly the equations of any asymptotes and the coordinates of any points of intersections with the coordinate axes.

The region bounded by the $x$-axis, the curve $y=f\left(x\right)$, and the lines $x=5$ and $x=7$ is rotated through $2\mathrm{\pi }$ about the $x$-axis. Find the volume of the solid generated, giving your answer in the form $\mathrm{\pi }(a+b \ln 2)$, where $a, b \in \mathrm{\mathbb{Z}}$.

State, in the context of the question, what the value of $34.50$ represents.

State, in the context of the question, what the value of $8.50$ represents.

Write down the minimum number of pizzas that can be ordered.

Kaelani has $450 \text{PGK}$.

Find the maximum number of large cheese pizzas that Kaelani can order from Olava’s Pizza Company.

State the range of $f$.

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

Find the coordinates of the point(s) where the graphs of $y=f\left(x\right)$ and $y=f^{-1}\left(x\right)$ intersect.

The line $L$ is tangent to the graphs of both $f$ and the inverse function $f^{-1}$.

Find the shaded area enclosed by the graphs of $f$ and $f^{-1}$ and the line $L$.

Find the range of $f$.

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

The function $g$ is defined by $g\left(x\right)=\frac{ax+4}{3-x}$, where $x\in \mathrm{ℝ}, x\ne 3$ and $a\in \mathrm{ℝ}$.

Given that $g\left(x\right)=g^{-1}\left(x\right)$, determine the value of $a$.

the vertical asymptote of the graph of $f$.

the $y$-axis.

the $x$-axis.

Sketch the graph of $f$ on the axes below.

the horizontal asymptote of the graph of $f$.

Find $g\left(0\right)$.

Find the value of $x$ such that $f\left(x\right)=0$.

The range of $f$ is $a\le y\le b$, where $a, b\in \mathrm{ℝ}$.

Find the value of $a$ and the value of $b$.

Sketch the curve $y=f\left(x\right)$, clearly indicating any asymptotes with their equations. State the coordinates of any local maximum or minimum points and any points of intersection with the coordinate axes.

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

Plant $A$ correct to three significant figures.

Plant $B$.

Find the largest possible domain $D$ for $f$ to be a function.

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

Explain why $f$ is an even function.

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

Find the inverse function $g^{-1}$ and state its domain.

Find $g^{\prime }(x)$.

Hence, show that there are no solutions to $g^{\prime }(x)=0$;

Hence, show that there are no solutions to $(g^{-1}{)}^{\prime }(x)=0$.

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.

Write down the range of $f$.

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

Write down the domain and range of $f^{-1}$.

This region is now rotated through $2\pi$ radians about the $x$-axis. Find the volume of revolution.

Sketch the graph of $f$ indicating clearly any intercepts with the axes and the coordinates of any local maximum or minimum points.

State the range of $f$.

Solve the inequality $\left|3x\arccos (x)\right|>1$.

On the grid, sketch the graph of $f^{-1}$.

Write down the range of $f$.

Write down the domain of $g$.

Find the range of f.

Write down the x-coordinate of P and the x-coordinate of Q.

Write down the values of x for which $f\left(x\right)>g\left(x\right)$.

Draw the line $y=-6$ on the axes.

Write down the number of solutions to $f(x)=-6$.

Find the range of values of $k$ for which $f(x)=k$ has no solution.

Find the value of $f\left(x\right)$ when $x=\frac{1}{2}$.

Find the function, $P\left(x\right)$ , that would define this footpath on the map.

State the domain of $P$.

Find the coordinates of the bridges relative to the centre of Orangeton.

Find the distance from the centre of Orangeton to the point at which the road meets the highway.

This straight road crosses the highway and then carries on due north.

State whether the straight road will ever cross the river. Justify your answer.

Write down the $y$-intercept of the graph of $y=f\left(x\right)$.

Sketch the graph of $y=f\left(x\right)$ for −3 ≤ $x$ ≤ 3 and −4 ≤ $y$ ≤ 12.

Determine the range of $f\left(x\right)$ for $p$ ≤ $x$ ≤ $q$.

Write down the equation of the axis of symmetry for this graph.

Find the value of $b$.

Write down the range of $f(x)$.

Find the value of $\left(f\circ f\right)\left(1\right)$.

Given that $f^{-1}\left(a\right)=3$, determine the value of $a$.

Given that $g\left(x\right)=2f\left(x-1\right)$, find the domain and range of $g$.

Write down the coordinates of $\text{B}$.

Given that $f^{\prime }\left(a\right)=\frac{1}{\text{ln}p}$, find the equation of $L_1$ in terms of $x$, $p$ and $q$.

The line $L_2$ is tangent to the graph of $g$ at $\text{A}$ and has equation $y=\left(\text{ln}p\right)x+q+1$.

The line $L_2$ passes through the point $\left(-2\text{, }-2\right)$.

The gradient of the normal to $g$ at $\text{A}$ is $\frac{1}{\text{ln}\left(\frac{1}{3}\right)}$.

Find the equation of $L_1$ in terms of $x$.

Write down the value of $k$.