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

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

8 The functions f and g are defined as follows, where \(a\) and \(b\) are constants. $$\begin{aligned} & \mathrm { f } ( x ) = 1 + \frac { 2 a } { x - a } \text { for } x > a \\ & \mathrm {~g} ( x ) = b x - 2 \text { for } x \in \mathbb { R } \end{aligned}$$

1. Given that \(\mathrm { f } ( 7 ) = \frac { 5 } { 2 }\) and \(\mathrm { gf } ( 5 ) = 4\), find the values of \(a\) and \(b\).
   For the rest of this question, you should use the value of \(a\) which you found in (a).
2. Find the domain of \(\mathrm { f } ^ { - 1 }\).
3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).

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

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

1. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
2. Show that \(\frac { 2 } { 3 } + \frac { 2 } { 3 ( 3 x - 1 ) }\) can be expressed as \(\frac { 2 x } { 3 x - 1 }\).
3. State the range of f.

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

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

8 The function f is defined by \(\mathrm { f } ( x ) = 2 - \frac { 3 } { 4 x - p }\) for \(x > \frac { p } { 4 }\), where \(p\) is a constant.

1. Find \(\mathrm { f } ^ { \prime } ( x )\) and hence determine whether f is an increasing function, a decreasing function or neither.
2. Express \(\mathrm { f } ^ { - 1 } ( x )\) in the form \(\frac { p } { a } - \frac { b } { c x - d }\), where \(a , b , c\) and \(d\) are integers.
3. Hence state the value of \(p\) for which \(\mathrm { f } ^ { - 1 } ( x ) \equiv \mathrm { f } ( x )\).

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 \(A P B = \frac { 5 } { 6 } \pi\).

2 The function f is defined by \(\mathrm { f } ( x ) = - 2 x ^ { 2 } - 8 x - 13\) for \(x < - 3\).

1. Express \(\mathrm { f } ( x )\) in the form \(- 2 ( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are integers.
2. Find the range of f.
3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).

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