1.02u Functions: definition and vocabulary (domain, range, mapping)

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 \(\mathrm { h } ^ { - 1 }\).

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 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.

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 \(\mathrm { fg } ( x )\).

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 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 \(b\).

10

1. Express \(x ^ { 2 } - 2 x - 15\) in the form \(( x + a ) ^ { 2 } + b\). The function f is defined for \(p \leqslant x \leqslant q\), where \(p\) and \(q\) are positive constants, by $$f : x \mapsto x ^ { 2 } - 2 x - 15$$ The range of f is given by \(c \leqslant \mathrm { f } ( x ) \leqslant d\), where \(c\) and \(d\) are constants.
2. State the smallest possible value of \(c\). For the case where \(c = 9\) and \(d = 65\),
3. find \(p\) and \(q\),
4. find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).

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.

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 \(\mathrm { g } ^ { - 1 } ( x )\).

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).

5. \begin{figure}[h] \end{figure} Figure 1 shows the sketch of a curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The curve crosses the \(y\)-axis at \(( 0,4 )\) and crosses the \(x\)-axis at \(( 5,0 )\). The curve has a single turning point, a maximum, at (2, 7). The line with equation \(y = 1\) is the only asymptote to the curve.

1. State the coordinates of the turning point on the curve with equation \(y = \mathrm { f } ( x - 2 )\).
2. State the solution of the equation f( \(2 x\) ) \(= 0\)
3. State the equation of the asymptote to the curve with equation \(y = \mathrm { f } ( - x )\). Given that the line with equation \(y = k\), where \(k\) is a constant, meets the curve \(y = \mathrm { f } ( x )\) at only one point,
4. state the set of possible values for \(k\).

9. $$\mathrm { f } ( \theta ) = 5 \cos \theta - 4 \sin \theta \quad \theta \in \mathbb { R }$$

1. Express \(\mathrm { f } ( \theta )\) in the form \(R \cos ( \theta + \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the exact value of \(R\) and give the value of \(\alpha\), in radians, to 3 decimal places. The curve with equation \(y = \cos \theta\) is transformed onto the curve with equation \(y = \mathrm { f } ( \theta )\) by a sequence of two transformations. Given that the first transformation is a stretch and the second a translation,
2. 1. describe fully the transformation that is a stretch,
   2. describe fully the transformation that is a translation. Given $$g ( \theta ) = \frac { 90 } { 4 + ( f ( \theta ) ) ^ { 2 } } \quad \theta \in \mathbb { R }$$
3. find the range of g.

   Leave blank	Q9
   END

6. The function f is defined by $$f ( x ) = \frac { 5 x - 3 } { x - 4 } \quad x > 4$$

1. Show, by using calculus, that f is a decreasing function.
2. Find \(\mathrm { f } ^ { - 1 }\)
3. 1. Show that \(\mathrm { ff } ( x ) = \frac { a x + b } { x + c }\) where \(a , b\) and \(c\) are constants to be found.
   2. Deduce the range of ff.

1. The functions f and g are defined by

$$\begin{array} { l l l } \mathrm { f } ( x ) = 9 - x ^ { 2 } & x \in \mathbb { R } & x \geqslant 0 \\ \mathrm {~g} ( x ) = \frac { 3 } { 2 x + 1 } & x \in \mathbb { R } & x \geqslant 0 \end{array}$$

1. Write down the range of f
2. Find the value of fg(1.5)
3. Find \(\mathrm { g } ^ { - 1 }\)

4. The functions f and g are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 4 x + 6 } { x - 5 } & x \in \mathbb { R } , x \neq 5 \\ \mathrm {~g} ( x ) = 5 - 2 x ^ { 2 } & x \in \mathbb { R } , x \leqslant 0 \end{array}$$

1. Solve the equation $$\operatorname { fg } ( x ) = 3$$
2. Find \(\mathrm { f } ^ { - 1 }\)
3. Sketch and label, on the same axes, the curve with equation \(y = \mathrm { g } ( x )\) and the curve with equation \(y = \mathrm { g } ^ { - 1 } ( x )\). Show on your sketch the coordinates of the points where each curve meets or cuts the coordinate axes.

2. The functions f and g are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 5 - x } { 3 x + 2 } & x \in \mathbb { R } , x \neq - \frac { 2 } { 3 } \\ \mathrm {~g} ( x ) = 2 x - 7 & x \in \mathbb { R } \end{array}$$

1. Find the value of \(\mathrm { fg } ( 5 )\)
2. Find \(\mathrm { f } ^ { - 1 }\)
3. Solve the equation $$\mathrm { f } \left( \frac { 1 } { a } \right) = \mathrm { g } ( a + 3 )$$

1. The function f is defined by

$$\mathrm { f } ( x ) = 2 x ^ { 2 } - 5 \quad x \geqslant 0 \quad x \in \mathbb { R }$$

1. State the range of f On the following page there is a diagram, labelled Diagram 1, which shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\).
2. On Diagram 1, sketch the curve with equation \(y = \mathrm { f } ^ { - 1 } ( x )\). The curve with equation \(y = \mathrm { f } ( x )\) meets the curve with equation \(y = \mathrm { f } ^ { - 1 } ( x )\) at the point \(P\) Using algebra and showing your working,
3. find the exact \(x\) coordinate of \(P\)
   \includegraphics[max width=\textwidth, alt={}]{bef290fb-fbac-4c9c-981e-5e323ac7182e-09_607_610_248_731}
   \section*{Diagram 1}

1. The functions \(f\) and \(g\) are defined by

$$\begin{aligned} & \mathrm { f } ( x ) = 2 + 5 \ln x \quad x > 0 \\ & \mathrm {~g} ( x ) = \frac { 6 x - 2 } { 2 x + 1 } \quad x > \frac { 1 } { 3 } \end{aligned}$$

1. Find \(\mathrm { f } ^ { - 1 } ( 22 )\)
2. Use differentiation to prove that g is an increasing function.
3. Find \(\mathrm { g } ^ { - 1 }\)
4. Find the range of fg