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

11 Functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto 2 x ^ { 2 } - 8 x + 10 & \text { for } 0 \leqslant x \leqslant 2 \\ \mathrm {~g} : x \mapsto x & \text { for } 0 \leqslant x \leqslant 10 \end{array}$$

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. State the domain of \(\mathrm { f } ^ { - 1 }\).
4. Sketch on the same diagram the graphs of \(y = \mathrm { f } ( x ) , y = \mathrm { g } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), making clear the relationship between the graphs.
5. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).

2 The functions f and g are defined for \(x \in \mathbb { R }\) by $$\begin{aligned} & \mathrm { f } : x \mapsto 3 x + a \\ & \mathrm {~g} : x \mapsto b - 2 x \end{aligned}$$ where \(a\) and \(b\) are constants. Given that \(\mathrm { ff } ( 2 ) = 10\) and \(\mathrm { g } ^ { - 1 } ( 2 ) = 3\), find

1. the values of \(a\) and \(b\),
2. an expression for \(\mathrm { fg } ( x )\).

10 The function f is defined by \(\mathrm { f } ( x ) = 4 x ^ { 2 } - 24 x + 11\), for \(x \in \mathbb { R }\).

1. Express \(\mathrm { f } ( x )\) in the form \(a ( x - b ) ^ { 2 } + c\) and hence state the coordinates of the vertex of the graph of \(y = \mathrm { f } ( x )\). The function g is defined by \(\mathrm { g } ( x ) = 4 x ^ { 2 } - 24 x + 11\), for \(x \leqslant 1\).
2. State the range of g .
3. Find an expression for \(\mathrm { g } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { g } ^ { - 1 }\).

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

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

6 The functions f and g are defined for \(- \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi\) by $$\begin{aligned} & f ( x ) = \frac { 1 } { 2 } x + \frac { 1 } { 6 } \pi \\ & g ( x ) = \cos x \end{aligned}$$ Solve the following equations for \(- \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi\).

1. \(\operatorname { gf } ( x ) = 1\), giving your answer in terms of \(\pi\).
2. \(\operatorname { fg } ( x ) = 1\), giving your answers correct to 2 decimal places.

7 \includegraphics[max width=\textwidth, alt={}, center]{d3c76ceb-cff7-4155-9697-5c302a9d63a9-3_821_688_255_731}

1. The diagram shows part of the curve \(y = 11 - x ^ { 2 }\) and part of the straight line \(y = 5 - x\) meeting at the point \(A ( p , q )\), where \(p\) and \(q\) are positive constants. Find the values of \(p\) and \(q\).
2. The function f is defined for the domain \(x \geqslant 0\) by $$f ( x ) = \begin{cases} 11 - x ^ { 2 } & \text { for } 0 \leqslant x \leqslant p \\ 5 - x & \text { for } x > p \end{cases}$$ Express \(\mathrm { f } ^ { - 1 } ( x )\) in a similar way.

5 The function f is defined by $$\mathrm { f } : x \mapsto x ^ { 2 } + 1 \text { for } x \geqslant 0$$

1. Define in a similar way the inverse function \(\mathrm { f } ^ { - 1 }\).
2. Solve the equation \(\operatorname { ff } ( x ) = \frac { 185 } { 16 }\).

8 A function f is defined by \(\mathrm { f } : x \mapsto 3 \cos x - 2\) for \(0 \leqslant x \leqslant 2 \pi\).

1. Solve the equation \(\mathrm { f } ( x ) = 0\).
2. Find the range of f .
3. Sketch the graph of \(y = \mathrm { f } ( x )\). A function g is defined by \(\mathrm { g } : x \mapsto 3 \cos x - 2\) for \(0 \leqslant x \leqslant k\).
4. State the maximum value of \(k\) for which g has an inverse.
5. Obtain an expression for \(\mathrm { g } ^ { - 1 } ( x )\).

10 A curve has equation \(y = 2 x ^ { 2 } - 3 x\).

1. Find the set of values of \(x\) for which \(y > 9\).
2. Express \(2 x ^ { 2 } - 3 x\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants, and state the coordinates of the vertex of the curve. The functions f and g are defined for all real values of \(x\) by $$\mathrm { f } ( x ) = 2 x ^ { 2 } - 3 x \quad \text { and } \quad \mathrm { g } ( x ) = 3 x + k$$ where \(k\) is a constant.
3. Find the value of \(k\) for which the equation \(\mathrm { gf } ( x ) = 0\) has equal roots.

10 The function f is defined by \(\mathrm { f } : x \mapsto x ^ { 2 } + 4 x\) for \(x \geqslant c\), where \(c\) is a constant. It is given that f is a one-one function.

1. State the range of f in terms of \(c\) and find the smallest possible value of \(c\). The function g is defined by \(\mathrm { g } : x \mapsto a x + b\) for \(x \geqslant 0\), where \(a\) and \(b\) are positive constants. It is given that, when \(c = 0 , \operatorname { gf } ( 1 ) = 11\) and \(\operatorname { fg } ( 1 ) = 21\).
2. Write down two equations in \(a\) and \(b\) and solve them to find the values of \(a\) and \(b\).

9

1. Express \(- x ^ { 2 } + 6 x - 5\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants. The function \(\mathrm { f } : x \mapsto - x ^ { 2 } + 6 x - 5\) is defined for \(x \geqslant m\), where \(m\) is a constant.
2. State the smallest value of \(m\) for which f is one-one.
3. For the case where \(m = 5\), find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
   [0pt] [Questions 10 and 11 are printed on the next page.] {www.cie.org.uk} after the live examination series. } \includegraphics[max width=\textwidth, alt={}, center]{a9e04003-1e43-40c4-991a-36aa3a93654b-4_773_641_260_753} The diagram shows a cuboid \(O A B C P Q R S\) with a horizontal base \(O A B C\) in which \(A B = 6 \mathrm {~cm}\) and \(O A = a \mathrm {~cm}\), where \(a\) is a constant. The height \(O P\) of the cuboid is 10 cm . The point \(T\) on \(B R\) is such that \(B T = 8 \mathrm {~cm}\), and \(M\) is the mid-point of \(A T\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O P\) respectively.

1 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 \(\mathrm { f } ^ { - 1 } ( x ) = \operatorname { gf } ( x )\).

8 The function f is defined by \(\mathrm { f } ( x ) = 3 x + 1\) for \(x \leqslant a\), where \(a\) is a constant. The function g is defined by \(\mathrm { g } ( x ) = - 1 - x ^ { 2 }\) for \(x \leqslant - 1\).

1. Find the largest value of \(a\) for which the composite function gf can be formed. For the case where \(a = - 1\),
2. solve the equation \(\operatorname { fg } ( x ) + 14 = 0\),
3. find the set of values of \(x\) which satisfy the inequality \(\operatorname { gf } ( x ) \leqslant - 50\).

8 The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } ( x ) = \frac { 4 } { x } - 2 \quad \text { for } x > 0 \\ & \mathrm {~g} ( x ) = \frac { 4 } { 5 x + 2 } \quad \text { for } x \geqslant 0 \end{aligned}$$

1. Find and simplify an expression for \(\mathrm { fg } ( x )\) and state the range of fg.
2. Find an expression for \(\mathrm { g } ^ { - 1 } ( x )\) and find the domain of \(\mathrm { g } ^ { - 1 }\).

10 A function f is defined by \(\mathrm { f } : x \mapsto 5 - 2 \sin 2 x\) for \(0 \leqslant x \leqslant \pi\).

1. Find the range of f .
2. Sketch the graph of \(y = \mathrm { f } ( x )\).
3. Solve the equation \(\mathrm { f } ( x ) = 6\), giving answers in terms of \(\pi\). The function g is defined by \(\mathrm { g } : x \mapsto 5 - 2 \sin 2 x\) for \(0 \leqslant x \leqslant k\), where \(k\) is a constant.
4. State the largest value of \(k\) for which g has an inverse.
5. For this value of \(k\), find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).

9 Functions f and g are defined for \(x > 3\) by $$\begin{aligned} & \mathrm { f } : x \mapsto \frac { 1 } { x ^ { 2 } - 9 } \\ & \mathrm {~g} : x \mapsto 2 x - 3 \end{aligned}$$

1. Find and simplify an expression for \(\operatorname { gg } ( x )\).
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
3. Solve the equation \(\operatorname { fg } ( x ) = \frac { 1 } { 7 }\).

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

1. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and find the point of intersection of the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\).
2. Sketch, on the same diagram, the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), making clear the relationship between the graphs.

6

1. The function f , defined by \(\mathrm { f } : x \mapsto a + b \sin x\) for \(x \in \mathbb { R }\), is such that \(\mathrm { f } \left( \frac { 1 } { 6 } \pi \right) = 4\) and \(\mathrm { f } \left( \frac { 1 } { 2 } \pi \right) = 3\).
   1. Find the values of the constants \(a\) and \(b\).
   2. Evaluate \(\mathrm { ff } ( 0 )\).
2. The function g is defined by \(\mathrm { g } : x \mapsto c + d \sin x\) for \(x \in \mathbb { R }\). The range of g is given by \(- 4 \leqslant \mathrm {~g} ( x ) \leqslant 10\). Find the values of the constants \(c\) and \(d\).

6 The functions f and g are defined by $$\begin{aligned} & \mathrm { f } ( x ) = \frac { 2 } { x ^ { 2 } - 1 } \text { for } x < - 1 \\ & \mathrm {~g} ( x ) = x ^ { 2 } + 1 \text { for } x > 0 \end{aligned}$$

1. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
2. Solve the equation \(\operatorname { gf } ( x ) = 5\).

11

1. The one-one function f is defined by \(\mathrm { f } ( x ) = ( x - 3 ) ^ { 2 } - 1\) for \(x < a\), where \(a\) is a constant.
   1. State the greatest possible value of \(a\).
   2. It is given that \(a\) takes this greatest possible value. State the range of f and find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
2. The function g is defined by \(\mathrm { g } ( x ) = ( x - 3 ) ^ { 2 }\) for \(x \geqslant 0\).
   1. Show that \(\operatorname { gg } ( 2 x )\) can be expressed in the form \(( 2 x - 3 ) ^ { 4 } + b ( 2 x - 3 ) ^ { 2 } + c\), where \(b\) and \(c\) are constants to be found.
   2. Hence expand \(\operatorname { gg } ( 2 x )\) completely, simplifying your answer.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

11

1. Express \(2 x ^ { 2 } - 12 x + 11\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
   The function f is defined by \(\mathrm { f } ( x ) = 2 x ^ { 2 } - 12 x + 11\) for \(x \leqslant k\).
2. State the largest value of the constant \(k\) for which f is a one-one function.
3. For this value of \(k\) find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
   The function g is defined by \(\mathrm { g } ( x ) = x + 3\) for \(x \leqslant p\).
4. With \(k\) now taking the value 1 , find the largest value of the constant \(p\) which allows the composite function fg to be formed, and find an expression for \(\mathrm { fg } ( x )\) whenever this composite function exists.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto \frac { 3 } { 2 x + 1 } \quad \text { for } x > 0 \\ & \mathrm {~g} : x \mapsto \frac { 1 } { x } + 2 \quad \text { for } x > 0 \end{aligned}$$

1. Find the range of f and the range of g .
2. Find an expression for \(\mathrm { fg } ( x )\), giving your answer in the form \(\frac { a x } { b x + c }\), where \(a , b\) and \(c\) are integers.
3. Find an expression for \(( \mathrm { fg } ) ^ { - 1 } ( x )\), giving your answer in the same form as for part (ii).

9 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } ( x ) = 2 x ^ { 2 } + 8 x + 1 \quad \text { for } x \in \mathbb { R } \\ & \mathrm {~g} ( x ) = 2 x - k \quad \text { for } x \in \mathbb { R } \end{aligned}$$ where \(k\) is a constant.

1. Find the value of \(k\) for which the line \(y = \mathrm { g } ( x )\) is a tangent to the curve \(y = \mathrm { f } ( x )\).
2. In the case where \(k = - 9\), find the set of values of \(x\) for which \(\mathrm { f } ( x ) < \mathrm { g } ( x )\).
3. In the case where \(k = - 1\), find \(\mathrm { g } ^ { - 1 } \mathrm { f } ( x )\) and solve the equation \(\mathrm { g } ^ { - 1 } \mathrm { f } ( x ) = 0\).
4. Express \(\mathrm { f } ( x )\) in the form \(2 ( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants, and hence state the least value of \(\mathrm { f } ( x )\).

2 The function g is defined by \(\mathrm { g } ( x ) = x ^ { 2 } - 6 x + 7\) for \(x > 4\). By first completing the square, find an expression for \(\mathrm { g } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { g } ^ { - 1 }\).

9

1. Express \(- x ^ { 2 } + 6 x - 5\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
   The function \(\mathrm { f } : x \mapsto - x ^ { 2 } + 6 x - 5\) is defined for \(x \geqslant m\), where \(m\) is a constant.
2. State the smallest value of \(m\) for which f is one-one.
3. For the case where \(m = 5\), find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{097c5d00-9f92-4c3e-8056-7de09347fbb6-16_771_636_260_756} The diagram shows a cuboid \(O A B C P Q R S\) with a horizontal base \(O A B C\) in which \(A B = 6 \mathrm {~cm}\) and \(O A = a \mathrm {~cm}\), where \(a\) is a constant. The height \(O P\) of the cuboid is 10 cm . The point \(T\) on \(B R\) is such that \(B T = 8 \mathrm {~cm}\), and \(M\) is the mid-point of \(A T\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O P\) respectively.