Functions

Show that $\mathrm{gf}(x) = 8x^2 - k - 1$.

The curve with equation $y = 8x^2 - k - 1$ is transformed to the curve with equation $y = \mathrm{h}(x)$ by the following sequence of transformations.

Find an expression for $\mathrm{h}(x)$ in terms of $x$ and $k$.

The range of $\mathrm{h}$ is given by $\mathrm{h}(x) \leqslant 15$.

Find the values of $c$ and $k$.

Describe fully a sequence of transformations which transforms the graph of $y = f(x)$ to the graph of $y = g(x)$. You should make clear the order in which the transformations are applied.

The diagram shows the graph of $y = g(x)$.

On the diagram sketch the graph of $y = g^{-1}(x)$ together with any relevant mirror line.

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

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

The function $h$ is defined by

Find the value of $g^{-1}h(4)$.

Explain why the composite function $hg^{-1}$ cannot be formed.

Describe fully the transformations that have been combined to transform the graph of $y = \mathrm{f}(x)$ to the graph of $y = \mathrm{g}(x)$.

On the given axes, sketch the graphs of $y = \mathrm{f}(x)$ and $y = \mathrm{g}(x)$.

Find $\mathrm{g}^{-1}\mathrm{f}\left(\frac{1}{3}\pi\right)$.

Explain why the composite function $\mathrm{fg}$ cannot be formed.

State the range of $f$.

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

Solve the equation $gf(x) = 69$.

The graph of $y = f(x)$ is transformed to the graph of $y = f(3x) + 2$.

Describe fully the two transformations which have been combined to give the resulting graph.

A different graph has equation $y = g(x)$. This graph is stretched by scale factor 3 in the $y$-direction and then reflected in the $y$-axis.

Write down the equation of the transformed graph in terms of the function $g$.

Describe fully a suitable sequence of transformations. Make clear the order in which the transformations are applied.

You are given that $f(x) = a(x + b)^3 + c$.

State the values of the constants $a$, $b$ and $c$.

It is given that $f(a) = 4$.

Find the value of $a$.

Find an expression for $f^{-1}(x)$ and state the domain of $f^{-1}$.

The function $g$ is defined by

for $x > \frac{3}{2}$.

Show that $fg(x) \equiv kx$, where $k$ is a constant to be determined.

Given that the range of the function $gf$ is $gf(x) \ge 39$, find the value of $k$.

For this value of $k$, determine the range of the function $fg$.

The function $h$ is defined for all real values of $x$ and is such that $gh(x) = 35x + 19$.

Find an expression for $g^{-1}(x)$ and hence, or otherwise, find an expression for $h(x)$.