[Maximum number: 4]

The graph of y=f(x) is transformed to the graph of y=3-f(x).
Describe fully, in the correct order, the two transformations that have been combined.

(a)

The curve with equation $y=x^{2}$ is transformed to the curve with equation $y=x^{2}+6 x+5$ .

Describe fully the transformation(s) involved.

[ 2 ]

Functions f and g are defined by

\begin{aligned} & \mathrm{f}: x \mapsto 10-3 x, \quad x \in \mathbb{R}, \\ & \mathrm{~g}: x \mapsto \frac{10}{3-2 x}, \quad x \in \mathbb{R}, x \neq \frac{3}{2} \end{aligned}

Solve the equation ff(x)=gf(2).

[Maximum number: 4]

Functions f and g are defined by

\begin{aligned} & \mathrm{f}: x \mapsto 3 x+2, \quad x \in \mathbb{R}, \\ & \mathrm{~g}: x \mapsto 4 x-12, \quad x \in \mathbb{R} . \end{aligned}

Solve the equation $\mathrm{f}^{-1}(x)=\mathrm{gf}(x)$ .

[Maximum number: 5]

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

(a)

Describe fully the two single transformations that have been combined to give the resulting transformation.

[ 3 ]

(b)

The point P(5,6) lies on the transformed curve y=f(2 x)-3.

State the coordinates of the corresponding point on the original curve y=f(x).

[ 2 ]

[Maximum number: 4]

The graph of y=f(x) is transformed to the graph of $y=1+\mathrm{f}\left(\frac{1}{2} x\right)$ .
Describe fully the two single transformations which have been combined to give the resulting transformation.

[Maximum number: 6]

A function f is defined by f : $x \mapsto 4-5 x$ for $x \in \mathbb{R}$ .

(a)

Find an expression for $\mathrm{f}^{-1}(x)$ and find the point of intersection of the graphs of y=f(x) and $y=\mathrm{f}^{-1}(x)$ .

[ 3 ]

(b)

Sketch, on the same diagram, the graphs of y=f(x) and $y=\mathrm{f}^{-1}(x)$ , making clear the relationship between the graphs.

[ 3 ]

[Maximum number: 4]

A function f is such that $\mathrm{f}(x)=\sqrt{ }\left(\frac{x+3}{2}\right)+1$ , for $x \geqslant-3$ . Find

(a)

$\mathrm{f}^{-1}(x)$ in the form $a x^{2}+b x+c$ , where a, b and c are constants,

[ 3 ]

(b)

the domain of $f^{-1}$ .

[ 1 ]

[Maximum number: 4]

In each of parts (a), (b) and (c), the graph shown with solid lines has equation y=f(x). The graph shown with broken lines is a transformation of y=f(x).

(a)

State, in terms of f , the equation of the graph shown with broken lines.

[ 1 ]

(b)

State, in terms of f , the equation of the graph shown with broken lines.

[ 1 ]

(c)

State, in terms of f , the equation of the graph shown with broken lines.

[ 2 ]

Functions f and g are defined for $x \in \mathbb{R}$ by

\begin{aligned} & \mathrm{f}: x \mapsto 2 x+3 \\ & \mathrm{~g}: x \mapsto x^{2}-2 x \end{aligned}

Express $\operatorname{gf}(x)$ in the form $a(x+b)^{2}+c$ , where a, b and c are constants.