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

9 The functions f and g are defined by $$\begin{aligned} & \mathrm { f } ( x ) = x ^ { 2 } - 4 x + 3 \text { for } x > c , \text { where } c \text { is a constant, } \\ & \mathrm { g } ( x ) = \frac { 1 } { x + 1 } \quad \text { for } x > - 1 \end{aligned}$$

1. Express \(\mathrm { f } ( x )\) in the form \(( x - a ) ^ { 2 } + b\).
   It is given that f is a one-one function.
2. State the smallest possible value of \(c\).
   It is now given that \(c = 5\).
3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
4. Find an expression for \(\mathrm { gf } ( x )\) and state the range of gf .

9 Functions f and g are defined as follows: $$\begin{aligned} & \mathrm { f } ( x ) = ( x - 2 ) ^ { 2 } - 4 \text { for } x \geqslant 2 , \\ & \mathrm {~g} ( x ) = a x + 2 \text { for } x \in \mathbb { R } , \end{aligned}$$ where \(a\) is a constant.

1. State the range of f.
2. Find \(\mathrm { f } ^ { - 1 } ( x )\).
3. Given that \(a = - \frac { 5 } { 3 }\), solve the equation \(\mathrm { f } ( x ) = \mathrm { g } ( x )\).
4. Given instead that \(\operatorname { ggf } ^ { - 1 } ( 12 ) = 62\), find the possible values of \(a\).

5 The function f is defined by \(\mathrm { f } ( x ) = 2 x ^ { 2 } + 3\) for \(x \geqslant 0\).

1. Find and simplify an expression for \(\mathrm { ff } ( x )\).
2. Solve the equation \(\mathrm { ff } ( x ) = 34 x ^ { 2 } + 19\).

8 Functions f and g are defined as follows: $$\begin{aligned} & \mathrm { f } : x \mapsto x ^ { 2 } - 1 \text { for } x < 0 \\ & \mathrm {~g} : x \mapsto \frac { 1 } { 2 x + 1 } \text { for } x < - \frac { 1 } { 2 } \end{aligned}$$

1. Solve the equation \(\operatorname { fg } ( x ) = 3\).
2. Find an expression for \(( \mathrm { fg } ) ^ { - 1 } ( x )\).

6 The function \(f\) is defined as follows: $$\mathrm { f } ( x ) = \frac { x ^ { 2 } - 4 } { x ^ { 2 } + 4 } \quad \text { for } x > 2$$

1. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
2. Show that \(1 - \frac { 8 } { x ^ { 2 } + 4 }\) can be expressed as \(\frac { x ^ { 2 } - 4 } { x ^ { 2 } + 4 }\) and hence state the range of f .
3. Explain why the composite function ff cannot be formed.

10 Functions \(f\) and \(g\) are defined as follows: $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 2 x + 1 } { 2 x - 1 } & \text { for } x \neq \frac { 1 } { 2 } \\ \mathrm {~g} ( x ) = x ^ { 2 } + 4 & \text { for } x \in \mathbb { R } \end{array}$$

1. \includegraphics[max width=\textwidth, alt={}, center]{bb7595c9-93ae-49e8-9cc5-9ecc802e6060-16_773_1182_555_511} The diagram shows part of the graph of \(y = \mathrm { f } ( x )\).
   State the domain of \(\mathrm { f } ^ { - 1 }\).
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
3. Find \(\mathrm { gf } ^ { - 1 } ( 3 )\).
4. Explain why \(\mathrm { g } ^ { - 1 } ( x )\) cannot be found.
5. Show that \(1 + \frac { 2 } { 2 x - 1 }\) can be expressed as \(\frac { 2 x + 1 } { 2 x - 1 }\). Hence find the area of the triangle enclosed by the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where \(x = 1\) and the \(x\) - and \(y\)-axes.

6 The function f is defined by \(\mathrm { f } ( x ) = 2 x ^ { 2 } - 16 x + 23\) for \(x < 3\).

1. Express \(\mathrm { f } ( x )\) in the form \(2 ( x + a ) ^ { 2 } + b\).
2. Find the range of f.
3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
   The function g is defined by \(\mathrm { g } ( x ) = 2 x + 4\) for \(x < - 1\).
4. Find and simplify an expression for \(\mathrm { fg } ( x )\).

7 The function f is defined by \(\mathrm { f } ( x ) = 2 - \frac { 5 } { x + 2 }\) for \(x > - 2\).

1. 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 \(\mathrm { g } ( x ) = x + 3\) for \(x > 0\).
3. Obtain an expression for \(\operatorname { fg } ( x )\) giving your answer in the form \(\frac { a x + b } { c x + d }\), where \(a , b , c\) and \(d\) are integers.

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

1. Express \(2 x ^ { 2 } + 12 x + 11\) in the form \(2 ( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
   The function f is defined by \(\mathrm { f } ( x ) = 2 x ^ { 2 } + 12 x + 11\) for \(x \leqslant - 4\).
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
   The function g is defined by \(\mathrm { g } ( x ) = 2 x - 3\) for \(x \leqslant k\).
3. For the case where \(k = - 1\), solve the equation \(\operatorname { fg } ( x ) = 193\).
4. State the largest value of \(k\) possible for the composition fg to be defined.

7 Functions f and g are defined as follows: $$\begin{aligned} & \mathrm { f } : x \mapsto x ^ { 2 } + 2 x + 3 \text { for } x \leqslant - 1 , \\ & \mathrm {~g} : x \mapsto 2 x + 1 \text { for } x \geqslant - 1 . \end{aligned}$$

1. Express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b\) and state the range of f .
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
3. Solve the equation \(\operatorname { gf } ( x ) = 13\).

9 Functions f, g and h are defined as follows: $$\begin{aligned} & \mathrm { f } : x \mapsto x - 4 x ^ { \frac { 1 } { 2 } } + 1 \quad \text { for } x \geqslant 0 \\ & \mathrm {~g} : x \mapsto m x ^ { 2 } + n \quad \text { for } x \geqslant - 2 , \text { where } m \text { and } n \text { are constants, } \\ & \mathrm { h } : x \mapsto x ^ { \frac { 1 } { 2 } } - 2 \quad \text { for } x \geqslant 0 . \end{aligned}$$

1. Solve the equation \(\mathrm { f } ( x ) = 0\), giving your solutions in the form \(x = a + b \sqrt { c }\), where \(a , b\) and \(c\) are integers.
2. Given that \(\mathrm { f } ( x ) \equiv \mathrm { gh } ( x )\), find the values of \(m\) and \(n\). \includegraphics[max width=\textwidth, alt={}, center]{05e75fa2-81ae-44b1-b073-4100f5d911e0-16_652_1045_255_550} The diagram shows a circle with centre \(A\) of radius 5 cm and a circle with centre \(B\) of radius 8 cm . The circles touch at the point \(C\) so that \(A C B\) is a straight line. The tangent at the point \(D\) on the smaller circle intersects the larger circle at \(E\) and passes through \(B\).

9 The function f is defined by \(\mathrm { f } ( x ) = - 3 x ^ { 2 } + 2\) for \(x \leqslant - 1\).

1. State the range of f.
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
   The function g is defined by \(\mathrm { g } ( x ) = - x ^ { 2 } - 1\) for \(x \leqslant - 1\).
3. Solve the equation \(\mathrm { fg } ( x ) - \mathrm { gf } ( x ) + 8 = 0\).

9 The functions f and g are defined for all real values of \(x\) by $$f ( x ) = ( 3 x - 2 ) ^ { 2 } + k \quad \text { and } \quad g ( x ) = 5 x - 1$$ where \(k\) is a constant.

1. Given that the range of the function gf is \(\mathrm { gf } ( x ) \geqslant 39\), find the value of \(k\).
2. For this value of \(k\), determine the range of the function fg .
3. The function h is defined for all real values of \(x\) and is such that \(\mathrm { gh } ( x ) = 35 x + 19\). Find an expression for \(\mathrm { g } ^ { - 1 } ( x )\) and hence, or otherwise, find an expression for \(\mathrm { h } ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-12_739_625_260_721} The diagram shows the circle with centre \(C ( - 4,5 )\) and radius \(\sqrt { 20 }\) units. The circle intersects the \(y\)-axis at the points \(A\) and \(B\). The size of angle \(A C B\) is \(\theta\) radians.

11 The functions \(f\) and \(g\) are defined by $$\begin{array} { l l } f ( x ) = x ^ { 2 } + 3 & \text { for } x > 0 \\ g ( x ) = 2 x + 1 & \text { for } x > - \frac { 1 } { 2 } \end{array}$$

1. Find an expression for \(\mathrm { fg } ( x )\).
2. Find an expression for \(( \mathrm { fg } ) ^ { - 1 } ( x )\) and state the domain of \(( \mathrm { fg } ) ^ { - 1 }\).
3. Solve the equation \(\mathrm { fg } ( x ) - 3 = \mathrm { gf } ( x )\).

5 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } ( x ) = 4 x - 2 , \quad \text { for } x \in \mathbb { R } , \\ & \mathrm {~g} ( x ) = \frac { 4 } { x + 1 } , \quad \text { for } x \in \mathbb { R } , x \neq - 1 \end{aligned}$$

1. Find the value of fg (7).
2. Find the values of \(x\) for which \(\mathrm { f } ^ { - 1 } ( x ) = \mathrm { g } ^ { - 1 } ( x )\).

8

1. Express \(- 3 x ^ { 2 } + 12 x + 2\) in the form \(- 3 ( x - a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
   The one-one function f is defined by \(\mathrm { f } : x \mapsto - 3 x ^ { 2 } + 12 x + 2\) for \(x \leqslant k\).
2. State the largest possible value of the constant \(k\).
   It is now given that \(k = - 1\).
3. State the range of f.
4. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
   The result of translating the graph of \(y = \mathrm { f } ( x )\) by \(\binom { - 3 } { 1 }\) is the graph of \(y = \mathrm { g } ( x )\).
5. Express \(\mathrm { g } ( x )\) in the form \(p x ^ { 2 } + q x + r\), where \(p , q\) and \(r\) are constants.

3 The function f is defined as follows: $$\mathrm { f } ( x ) = \frac { x + 3 } { x - 1 } \text { for } x > 1$$

1. Find the value of \(\mathrm { ff } ( 5 )\).
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).

10 The function f is defined by \(\mathrm { f } ( x ) = x ^ { 2 } + \frac { k } { x } + 2\) for \(x > 0\).

1. Given that the curve with equation \(y = \mathrm { f } ( x )\) has a stationary point when \(x = 2\), find \(k\).
2. Determine the nature of the stationary point.
3. Given that this is the only stationary point of the curve, find the range of f .

6 \includegraphics[max width=\textwidth, alt={}, center]{cba7391d-2cf7-483c-9a21-bc9fd49860ee-08_608_597_258_772} The diagram shows the graph of \(y = \mathrm { f } ( x )\).

1. On this diagram sketch the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\). It is now given that \(\mathrm { f } ( x ) = - \frac { x } { \sqrt { 4 - x ^ { 2 } } }\) where \(- 2 < x < 2\).
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
   The function g is defined by \(\mathrm { g } ( x ) = 2 x\) for \(- a < x < a\), where \(a\) is a constant.
3. State the maximum possible value of \(a\) for which fg can be formed.
4. Assuming that fg can be formed, find and simplify an expression for \(\mathrm { fg } ( x )\).

9 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } ( x ) = x + \frac { 1 } { x } \quad \text { for } x > 0 \\ & \mathrm {~g} ( x ) = a x + 1 \quad \text { for } x \in \mathbb { R } \end{aligned}$$ where \(a\) is a constant.

1. Find an expression for \(\operatorname { gf } ( x )\).
2. Given that \(\operatorname { gf } ( 2 ) = 11\), find the value of \(a\).
3. Given that the graph of \(y = \mathrm { f } ( x )\) has a minimum point when \(x = 1\), explain whether or not f has an inverse.
   It is given instead that \(a = 5\).
4. Find and simplify an expression for \(\mathrm { g } ^ { - 1 } \mathrm { f } ( x )\).
5. Explain why the composite function fg cannot be formed. \includegraphics[max width=\textwidth, alt={}, center]{5d26c357-ea9f-47d9-8eca-2152901cf2f1-16_1143_1008_267_566} The diagram shows a cross-section RASB of the body of an aircraft. The cross-section consists of a sector \(O A R B\) of a circle of radius 2.5 m , with centre \(O\), a sector \(P A S B\) of another circle of radius 2.24 m with centre \(P\) and a quadrilateral \(O A P B\). Angle \(A O B = \frac { 2 } { 3 } \pi\) and angle \(A P B = \frac { 5 } { 6 } \pi\).

9

1. Express \(4 x ^ { 2 } - 12 x + 13\) in the form \(( 2 x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
   The function f is defined by \(\mathrm { f } ( x ) = 4 x ^ { 2 } - 12 x + 13\) for \(p < x < q\), where \(p\) and \(q\) are constants. The function g is defined by \(\mathrm { g } ( x ) = 3 x + 1\) for \(x < 8\).
2. Given that it is possible to form the composite function gf , find the least possible value of \(p\) and the greatest possible value of \(q\).
3. Find an expression for \(\operatorname { gf } ( x )\).
   The function h is defined by \(\mathrm { h } ( x ) = 4 x ^ { 2 } - 12 x + 13\) for \(x < 0\).
4. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).

7 The function f is defined by \(\mathrm { f } ( x ) = 1 + \frac { 3 } { x - 2 }\) for \(x > 2\).

1. 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 \(\mathrm { g } ( x ) = 2 x - 2\) for \(x > 0\).
3. Obtain a simplified expression for \(\mathrm { gf } ( x )\).

11 The function f is defined, for \(x \in \mathbb { R }\), by \(\mathrm { f } : x \mapsto x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants.

1. It is given that \(a = 6\) and \(b = - 8\). Find the range of f .
2. It is given instead that \(a = 5\) and that the roots of the equation \(\mathrm { f } ( x ) = 0\) are \(k\) and \(- 2 k\), where \(k\) is a constant. Find the values of \(b\) and \(k\).
3. Show that if the equation \(\mathrm { f } ( x + a ) = a\) has no real roots then \(a ^ { 2 } < 4 ( b - a )\).

6 The function f , where \(\mathrm { f } ( x ) = a \sin x + b\), is defined for the domain \(0 \leqslant x \leqslant 2 \pi\). Given that \(\mathrm { f } \left( \frac { 1 } { 2 } \pi \right) = 2\) and that \(\mathrm { f } \left( \frac { 3 } { 2 } \pi \right) = - 8\),

1. find the values of \(a\) and \(b\),
2. find the values of \(x\) for which \(\mathrm { f } ( x ) = 0\), giving your answers in radians correct to 2 decimal places,
3. sketch the graph of \(y = \mathrm { f } ( x )\).