Find intersection points

11 The function f is defined by \(\mathrm { f } ( x ) = 10 + 6 x - x ^ { 2 }\) for \(x \in \mathbb { R }\).

1. By completing the square, find the range of f . \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-16_2715_37_143_2010} The function g is defined by \(\mathrm { g } ( x ) = 4 x + k\) for \(x \in \mathbb { R }\) where \(k\) is a constant.
2. It is given that the graph of \(y = \mathrm { g } ^ { - 1 } \mathrm { f } ( x )\) meets the graph of \(y = \mathrm { g } ( x )\) at a single point \(P\). Determine the coordinates of \(P\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown. \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-18_2715_35_143_2012}

9 Functions f and g are defined for \(x \in \mathbb { R }\) by $$\begin{aligned} & \mathrm { f } : x \mapsto \frac { 1 } { 2 } x - 2 \\ & \mathrm {~g} : x \mapsto 4 + x - \frac { 1 } { 2 } x ^ { 2 } \end{aligned}$$

1. Find the points of intersection of the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\).
2. Find the set of values of \(x\) for which \(\mathrm { f } ( x ) > \mathrm { g } ( x )\).
3. Find an expression for \(\mathrm { fg } ( x )\) and deduce the range of fg .
   The function h is defined by \(\mathrm { h } : x \mapsto 4 + x - \frac { 1 } { 2 } x ^ { 2 }\) for \(x \geqslant k\).
4. Find the smallest value of \(k\) for which h has an inverse.

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the graph of \(y = \mathrm { g } ( x )\), where $$g ( x ) = 3 + \sqrt { x + 2 } , \quad x \geqslant - 2$$

1. State the range of g .
2. Find \(\mathrm { g } ^ { - 1 } ( x )\) and state its domain.
3. Find the exact value of \(x\) for which $$\mathrm { g } ( x ) = x$$
4. Hence state the value of \(a\) for which $$\mathrm { g } ( a ) = \mathrm { g } ^ { - 1 } ( a )$$

6 The function f is defined by $$\mathrm { f } : x \mapsto 1 + \sqrt { } x \quad \text { for } x \geqslant 0$$

1. State the domain and range of the inverse function \(\mathrm { f } ^ { - 1 }\).
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
3. By considering the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), show that the solution to the equation $$\mathrm { f } ( x ) = \mathrm { f } ^ { - 1 } ( x )$$ is \(x = \frac { 1 } { 2 } ( 3 + \sqrt { } 5 )\).

6 \includegraphics[max width=\textwidth, alt={}, center]{c940af95-e291-402a-856c-9090d13163d5-3_627_689_264_726} The function f is defined for all real values of \(x\) by $$f ( x ) = \sqrt [ 3 ] { \frac { 1 } { 2 } x + 2 }$$ The graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) meet at the point \(P\), and the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\) meets the \(x\)-axis at \(Q\) (see diagram).

1. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and determine the \(x\)-coordinate of the point \(Q\).
2. State how the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) are related geometrically, and hence show that the \(x\)-coordinate of the point \(P\) is the root of the equation $$x = \sqrt [ 3 ] { \frac { 1 } { 2 } x + 2 }$$
3. Use an iterative process, based on the equation \(x = \sqrt [ 3 ] { \frac { 1 } { 2 } x + 2 }\), to find the \(x\)-coordinate of \(P\), giving your answer correct to 2 decimal places.

7 \includegraphics[max width=\textwidth, alt={}, center]{71e01d8f-d0ed-4f17-b7cd-6f5a93bbe329-3_428_751_703_641} The diagram shows the curve \(y = \mathrm { f } ( x )\), where f is the function defined for all real values of \(x\) by $$\mathrm { f } ( x ) = 3 + 4 \mathrm { e } ^ { - x }$$

1. State the range of f .
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\), and state the domain and range of \(\mathrm { f } ^ { - 1 }\).
3. The straight line \(y = x\) meets the curve \(y = \mathrm { f } ( x )\) at the point \(P\). By using an iterative process based on the equation \(x = \mathrm { f } ( x )\), with a starting value of 3 , find the coordinates of the point \(P\). Show all your working and give each coordinate correct to 3 decimal places.
4. How is the point \(P\) related to the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) ?

3 The function f is defined by \(\mathrm { f } ( x ) = ( x - 3 ) ^ { 2 } - 17\) for \(x \geqslant k\), where \(k\) is a constant.

1. Given that \(\mathrm { f } ^ { - 1 } ( x )\) exists, state the least possible value of \(k\).
2. Evaluate \(\mathrm { ff } ( 5 )\).
3. Solve the equation \(\mathrm { f } ( x ) = x\).
4. Explain why your solution to part (c) is also the solution to the equation \(\mathrm { f } ( x ) = \mathrm { f } ^ { - 1 } ( x )\).

10

1. Sketch the graph of \(y = \frac { 1 } { x } + a\), where \(a\) is a positive constant.

8 The curves \(\mathrm { y } = \mathrm { h } ( \mathrm { x } )\) and \(\mathrm { y } = \mathrm { h } ^ { - 1 } ( \mathrm { x } )\), where \(\mathrm { h } ( x ) = x ^ { 3 } - 8\), are shown below.
The curve \(\mathrm { y } = \mathrm { h } ( \mathrm { x } )\) crosses the \(x\)-axis at B and the \(y\)-axis at A.
The curve \(\mathrm { y } = \mathrm { h } ^ { - 1 } ( \mathrm { x } )\) crosses the \(x\)-axis at D and the \(y\)-axis at C . \includegraphics[max width=\textwidth, alt={}, center]{c30a926b-d832-46f5-aa65-0066ef482c3d-7_826_819_520_255}

1. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).
2. Determine the coordinates of A, B, C and D.
3. Determine the equation of the perpendicular bisector of AB . Give your answer in the form \(\mathrm { y } = \mathrm { mx } + c\), where \(m\) and \(c\) are constants to be determined.
4. Points A , B , C and D lie on a circle. Determine the equation of the circle. Give your answer in the form \(( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }\), where \(a\), \(b\) and \(r ^ { 2 }\) are constants to be determined.

The functions f and g are defined for all real values of \(x\) by $$f(x) = 3x - 2 \quad \text{and} \quad g(x) = 3x + 7.$$ Find the exact coordinates of the point at which

1. the graph of \(y = fg(x)\) meets the \(x\)-axis, [3]
2. the graph of \(y = g(x)\) meets the graph of \(y = g^{-1}(x)\), [3]
3. the graph of \(y = |f(x)|\) meets the graph of \(y = |g(x)|\). [4]