DP Mathematics: Applications and Interpretation Questionbank

Find an expression for the inverse function$f^{-1}\left(x\right)$. The domain is not required.

Write down the range of $f^{-1}\left(x\right)$.

Find the range of $f$.

Find an expression for $f^{-1}\left(x\right)$. You are not required to state a domain.

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

Find the inverse function $f^{-1}$, stating its domain.

Express $g\left(x\right)$ in the form $A+\frac{B}{x-2}$ where A, B are constants.

Sketch the graph of $y=g\left(x\right)$. State the equations of any asymptotes and the coordinates of any intercepts with the axes.

The function $h$ is defined by $h\left(x\right)=\sqrt{x}$, for $x$ ≥ 0.

State the domain and range of $h\circ g$.

Sketch the graph of $y=f\left(x\right)$ for $-\frac{5\pi }{8}⩽x⩽\frac{\pi }{8}$.

With reference to your graph, explain why $f$ is a function on the given domain.

Explain why $f$ has no inverse on the given domain.

Explain why $f$ is not a function for $-\frac{3\pi }{4}⩽x⩽\frac{\pi }{4}$.

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

Sketch the graph of $y=g\left(t\right)$ for t ≤ 0. Give the coordinates of any intercepts and the equations of any asymptotes.

Find $\alpha$ and β in terms of $k$.

Show that $\alpha$ + β < −2.

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

Write down the least value of $a$ such that $g$ has an inverse.

For the value of $a$ found in part (b), write down the domain of $g^{-1}$.

For the value of $a$ found in part (b), find an expression for $g^{-1}\left(x\right)$.

Find $\left(f\circ g\right)\left(\left(x\text{,}y\right)\right)$.

Find $\left(g\circ f\right)\left(\left(x\text{,}y\right)\right)$.

State with a reason whether or not $f$ and $g$ commute.

Find the inverse of $f$.

Find $(g\circ f)(x)$.

Given that $\underset{x\to +\mathrm{\infty }}{lim}(g\circ f)(x)=-3$, find the value of $b$.

On the grid above, sketch the graph of f ⁻¹.

Show that $(f\circ g)(x)=x^4-4x^2+3$.

On the following grid, sketch the graph of $(f\circ g)(x)$, for $0⩽x⩽2.25$.

The equation $(f\circ g)(x)=k$ has exactly two solutions, for $0⩽x⩽2.25$. Find the possible values of $k$.

Find the gradient of $L$.

Find u.

Find the acute angle between $y=x$ and $L$.

Find $\left(f\circ f\right)\left(x\right)$.

Hence, write down $f^{-1}\left(x\right)$.

Hence or otherwise, find the obtuse angle formed by the tangent line to $f$ at $x=8$ and the tangent line to $f$ at $x=2$.

Find $f(8)$.

Find $(g\circ f)(x)$.

Solve $(g\circ f)(x)=0$.