1.02v Inverse and composite functions: graphs and conditions for existence

11 The function f is defined by \(\mathrm { f } : x \mapsto 6 x - x ^ { 2 } - 5\) for \(x \in \mathbb { R }\).

1. Find the set of values of \(x\) for which \(\mathrm { f } ( x ) \leqslant 3\).
2. Given that the line \(y = m x + c\) is a tangent to the curve \(y = \mathrm { f } ( x )\), show that \(4 c = m ^ { 2 } - 12 m + 16\). The function g is defined by \(\mathrm { g } : x \mapsto 6 x - x ^ { 2 } - 5\) for \(x \geqslant k\), where \(k\) is a constant.
3. Express \(6 x - x ^ { 2 } - 5\) in the form \(a - ( x - b ) ^ { 2 }\), where \(a\) and \(b\) are constants.
4. State the smallest value of \(k\) for which g has an inverse.
5. For this value of \(k\), find an expression for \(\mathrm { g } ^ { - 1 } ( x )\). {www.cie.org.uk} after the live examination series. }

10 The function f is such that \(\mathrm { f } ( x ) = 2 x + 3\) for \(x \geqslant 0\). The function g is such that \(\mathrm { g } ( x ) = a x ^ { 2 } + b\) for \(x \leqslant q\), where \(a , b\) and \(q\) are constants. The function fg is such that \(\operatorname { fg } ( x ) = 6 x ^ { 2 } - 21\) for \(x \leqslant q\).

1. Find the values of \(a\) and \(b\).
2. Find the greatest possible value of \(q\). It is now given that \(q = - 3\).
3. Find the range of fg.
4. Find an expression for \(( \mathrm { fg } ) ^ { - 1 } ( x )\) and state the domain of \(( \mathrm { fg } ) ^ { - 1 }\).

9 The function f is defined by \(\mathrm { f } : x \mapsto \frac { 2 } { 3 - 2 x }\) for \(x \in \mathbb { R } , x \neq \frac { 3 } { 2 }\).

1. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
   The function g is defined by \(\mathrm { g } : x \mapsto 4 x + a\) for \(x \in \mathbb { R }\), where \(a\) is a constant.
2. Find the value of \(a\) for which \(\operatorname { gf } ( - 1 ) = 3\).
3. Find the possible values of \(a\) given that the equation \(\mathrm { f } ^ { - 1 } ( x ) = \mathrm { g } ^ { - 1 } ( x )\) has two equal roots.

10 The function f is defined by \(\mathrm { f } ( x ) = 3 \tan \left( \frac { 1 } { 2 } x \right) - 2\), for \(- \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi\).

1. Solve the equation \(\mathrm { f } ( x ) + 4 = 0\), giving your answer correct to 1 decimal place.
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and find the domain of \(\mathrm { f } ^ { - 1 }\).
3. Sketch, on the same diagram, the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\).

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.

7 The function f is defined by \(\mathrm { f } : x \mapsto 7 - 2 x ^ { 2 } - 12 x\) for \(x \in \mathbb { R }\).

1. Express \(7 - 2 x ^ { 2 } - 12 x\) in the form \(a - 2 ( x + b ) ^ { 2 }\), where \(a\) and \(b\) are constants.
2. State the coordinates of the stationary point on the curve \(y = \mathrm { f } ( x )\).
   The function g is defined by \(\mathrm { g } : x \mapsto 7 - 2 x ^ { 2 } - 12 x\) for \(x \geqslant k\).
3. State the smallest value of \(k\) for which g has an inverse.
4. For this value of \(k\), find \(\mathrm { g } ^ { - 1 } ( x )\).

10 The one-one function f is defined by \(\mathrm { f } ( x ) = ( x - 2 ) ^ { 2 } + 2\) for \(x \geqslant c\), where \(c\) is a constant.

1. State the smallest possible value of \(c\).
   In parts (ii) and (iii) the value of \(c\) is 4 .
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
3. Solve the equation \(\mathrm { ff } ( x ) = 51\), giving your answer in the form \(a + \sqrt { } b\).

9 The function f is defined by \(\mathrm { f } ( x ) = 2 - 3 \cos x\) for \(0 \leqslant x \leqslant 2 \pi\).

1. State the range of f .
2. Sketch the graph of \(y = \mathrm { f } ( x )\). The function g is defined by \(\mathrm { g } ( x ) = 2 - 3 \cos x\) for \(0 \leqslant x \leqslant p\), where \(p\) is a constant.
3. State the largest value of \(p\) for which g has an inverse.
4. For this value of \(p\), find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).

7 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 3 x - 2 , \quad x \in \mathbb { R } , \\ & \mathrm {~g} : x \mapsto \frac { 2 x + 3 } { x - 1 } , \quad x \in \mathbb { R } , x \neq 1 \end{aligned}$$

1. Obtain expressions for \(\mathrm { f } ^ { - 1 } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\), stating the value of \(x\) for which \(\mathrm { g } ^ { - 1 } ( x )\) is not defined. [4]
2. Solve the equation \(\operatorname { fg } ( x ) = \frac { 7 } { 3 }\).

8 The function f is such that \(\mathrm { f } ( x ) = a ^ { 2 } x ^ { 2 } - a x + 3 b\) for \(x \leqslant \frac { 1 } { 2 a }\), where \(a\) and \(b\) are constants.

1. For the case where \(\mathrm { f } ( - 2 ) = 4 a ^ { 2 } - b + 8\) and \(\mathrm { f } ( - 3 ) = 7 a ^ { 2 } - b + 14\), find the possible values of \(a\) and \(b\).
2. For the case where \(a = 1\) and \(b = - 1\), find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and give the domain of \(\mathrm { f } ^ { - 1 }\).

8 The functions f and g are defined for \(x \geqslant 0\) by $$\begin{aligned} & \mathrm { f } : x \mapsto 2 x ^ { 2 } + 3 \\ & \mathrm {~g} : x \mapsto 3 x + 2 \end{aligned}$$

1. Show that \(\operatorname { gf } ( x ) = 6 x ^ { 2 } + 11\) and obtain an unsimplified expression for \(\operatorname { fg } ( x )\).
2. Find an expression for \(( \mathrm { fg } ) ^ { - 1 } ( x )\) and determine the domain of \(( \mathrm { fg } ) ^ { - 1 }\).
3. Solve the equation \(\mathrm { gf } ( 2 x ) = \mathrm { fg } ( x )\).

11

1. Express \(2 x ^ { 2 } + 8 x - 10\) in the form \(a ( x + b ) ^ { 2 } + c\).
2. For the curve \(y = 2 x ^ { 2 } + 8 x - 10\), state the least value of \(y\) and the corresponding value of \(x\).
3. Find the set of values of \(x\) for which \(y \geqslant 14\). Given that \(\mathrm { f } : x \mapsto 2 x ^ { 2 } + 8 x - 10\) for the domain \(x \geqslant k\),
4. find the least value of \(k\) for which f is one-one,
5. express \(\mathrm { f } ^ { - 1 } ( x )\) in terms of \(x\) in this case.

10 Functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 2 x - 5 , \quad x \in \mathbb { R } , \\ & \mathrm {~g} : x \mapsto \frac { 4 } { 2 - x } , \quad x \in \mathbb { R } , \quad x \neq 2 . \end{aligned}$$

1. Find the value of \(x\) for which \(\mathrm { fg } ( x ) = 7\).
2. Express each of \(\mathrm { f } ^ { - 1 } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\) in terms of \(x\).
3. Show that the equation \(\mathrm { f } ^ { - 1 } ( x ) = \mathrm { g } ^ { - 1 } ( x )\) has no real roots.
4. Sketch, on a single diagram, the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), making clear the relationship between these two graphs.

9 The function f : \(x \mapsto 2 x - a\), where \(a\) is a constant, is defined for all real \(x\).

1. In the case where \(a = 3\), solve the equation \(\mathrm { ff } ( x ) = 11\). The function \(\mathrm { g } : x \mapsto x ^ { 2 } - 6 x\) is defined for all real \(x\).
2. Find the value of \(a\) for which the equation \(\mathrm { f } ( x ) = \mathrm { g } ( x )\) has exactly one real solution. The function \(\mathrm { h } : x \mapsto x ^ { 2 } - 6 x\) is defined for the domain \(x \geqslant 3\).
3. Express \(x ^ { 2 } - 6 x\) in the form \(( x - p ) ^ { 2 } - q\), where \(p\) and \(q\) are constants.
4. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { h } ^ { - 1 }\).

8 A function f is defined by \(\mathrm { f } : x \mapsto ( 2 x - 3 ) ^ { 3 } - 8\), for \(2 \leqslant x \leqslant 4\).

1. Find an expression, in terms of \(x\), for \(\mathrm { f } ^ { \prime } ( x )\) and show that f is an increasing function.
2. Find an expression, in terms of \(x\), for \(\mathrm { f } ^ { - 1 } ( x )\) and find the domain of \(\mathrm { f } ^ { - 1 }\).

10 The function f is defined by \(\mathrm { f } : x \mapsto x ^ { 2 } - 3 x\) for \(x \in \mathbb { R }\).

1. Find the set of values of \(x\) for which \(\mathrm { f } ( x ) > 4\).
2. Express \(\mathrm { f } ( x )\) in the form \(( x - a ) ^ { 2 } - b\), stating the values of \(a\) and \(b\).
3. Write down the range of f .
4. State, with a reason, whether f has an inverse. The function g is defined by \(\mathrm { g } : x \mapsto x - 3 \sqrt { } x\) for \(x \geqslant 0\).
5. Solve the equation \(\mathrm { g } ( x ) = 10\).

11 The function f is defined by \(\mathrm { f } : x \mapsto 2 x ^ { 2 } - 8 x + 11\) for \(x \in \mathbb { R }\).

1. Express \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a\), \(b\) and \(c\) are constants.
2. State the range of f .
3. Explain why f does not have an inverse. The function g is defined by \(\mathrm { g } : x \mapsto 2 x ^ { 2 } - 8 x + 11\) for \(x \leqslant A\), where \(A\) is a constant.
4. State the largest value of \(A\) for which g has an inverse.
5. When \(A\) has this value, obtain an expression, in terms of \(x\), for \(\mathrm { g } ^ { - 1 } ( x )\) and state the range of \(\mathrm { g } ^ { - 1 }\).

10 The function f is defined by $$\mathrm { f } : x \mapsto 3 x - 2 \text { for } x \in \mathbb { R } .$$

1. Sketch, in a single diagram, the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), making clear the relationship between the two graphs. The function g is defined by $$\mathrm { g } : x \mapsto 6 x - x ^ { 2 } \text { for } x \in \mathbb { R }$$
2. Express \(\operatorname { gf } ( x )\) in terms of \(x\), and hence show that the maximum value of \(\operatorname { gf } ( x )\) is 9 . The function h is defined by $$\mathrm { h } : x \mapsto 6 x - x ^ { 2 } \text { for } x \geqslant 3$$
3. Express \(6 x - x ^ { 2 }\) in the form \(a - ( x - b ) ^ { 2 }\), where \(a\) and \(b\) are positive constants.
4. Express \(\mathrm { h } ^ { - 1 } ( x )\) in terms of \(x\).

9 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 2 \\ 3 \\ - 6 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 0 \\ - 6 \\ 8 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } - 2 \\ 5 \\ - 2 \end{array} \right)$$

1. Find angle \(A O B\).
2. Find the vector which is in the same direction as \(\overrightarrow { A C }\) and has magnitude 30 .
3. Find the value of the constant \(p\) for which \(\overrightarrow { O A } + p \overrightarrow { O B }\) is perpendicular to \(\overrightarrow { O C }\).

4 The function f is defined by f : \(x \mapsto 5 - 3 \sin 2 x\) for \(0 \leqslant x \leqslant \pi\).

1. Find the range of f .
2. Sketch the graph of \(y = \mathrm { f } ( x )\).
3. State, with a reason, whether f has an inverse.

7 \includegraphics[max width=\textwidth, alt={}, center]{b566719c-216e-41e5-8431-da77e1dad73e-3_301_485_264_829} A piece of wire of length 50 cm is bent to form the perimeter of a sector \(P O Q\) of a circle. The radius of the circle is \(r \mathrm {~cm}\) and the angle \(P O Q\) is \(\theta\) radians (see diagram).

1. Express \(\theta\) in terms of \(r\) and show that the area, \(A \mathrm {~cm} ^ { 2 }\), of the sector is given by $$A = 25 r - r ^ { 2 } .$$
2. Given that \(r\) can vary, find the stationary value of \(A\) and determine its nature.

2 In the expansion of \(( 1 + a x ) ^ { 6 }\), where \(a\) is a constant, the coefficient of \(x\) is - 30 . Find the coefficient of \(x ^ { 3 }\).

7 A function f is defined by f : \(x \mapsto 3 - 2 \tan \left( \frac { 1 } { 2 } x \right)\) for \(0 \leqslant x < \pi\).

1. State the range of f .
2. State the exact value of \(\mathrm { f } \left( \frac { 2 } { 3 } \pi \right)\).
3. Sketch the graph of \(y = \mathrm { f } ( x )\).
4. Obtain an expression, in terms of \(x\), for \(\mathrm { f } ^ { - 1 } ( x )\).

7 The function f is defined by $$\mathrm { f } ( x ) = x ^ { 2 } - 4 x + 7 \text { for } x > 2$$

1. Express \(\mathrm { f } ( x )\) in the form \(( x - a ) ^ { 2 } + b\) and hence state the range of f .
2. Obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\). The function g is defined by $$\mathrm { g } ( x ) = x - 2 \text { for } x > 2$$ The function h is such that \(\mathrm { f } = \mathrm { hg }\) and the domain of h is \(x > 0\).
3. Obtain an expression for \(\mathrm { h } ( x )\).

7 \includegraphics[max width=\textwidth, alt={}, center]{32a57386-2696-4fda-a3cb-ca0c5c3be432-3_778_816_255_662} The diagram shows the function f defined for \(0 \leqslant x \leqslant 6\) by $$\begin{aligned} & x \mapsto \frac { 1 } { 2 } x ^ { 2 } \quad \text { for } 0 \leqslant x \leqslant 2 , \\ & x \mapsto \frac { 1 } { 2 } x + 1 \text { for } 2 < x \leqslant 6 . \end{aligned}$$

1. State the range of f .
2. Copy the diagram and on your copy sketch the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\).
3. Obtain expressions to define \(\mathrm { f } ^ { - 1 } ( x )\), giving the set of values of \(x\) for which each expression is valid.