Find inverse function

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 \(a\).

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.

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.

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 \(\mathrm { fg } ( x ) - \mathrm { gf } ( x ) + 8 = 0\).

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.

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

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

6 The function f is such that \(\mathrm { f } ( x ) = ( 3 x + 2 ) ^ { 3 } - 5\) for \(x \geqslant 0\).

1. Obtain an expression for \(\mathrm { f } ^ { \prime } ( x )\) and hence explain why f is an increasing function.
2. Obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).

10 Functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 3 x - 4 , \quad x \in \mathbb { R } , \\ & \mathrm {~g} : x \mapsto 2 ( x - 1 ) ^ { 3 } + 8 , \quad x > 1 . \end{aligned}$$

1. Evaluate fg(2).
2. Sketch in a single diagram the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), making clear the relationship between the graphs.
3. Obtain an expression for \(\mathrm { g } ^ { \prime } ( x )\) and use your answer to explain why g has an inverse.
4. Express each of \(\mathrm { f } ^ { - 1 } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\) in terms of \(x\).

9 A function f is defined by \(\mathrm { f } ( x ) = \frac { 5 } { 1 - 3 x }\), for \(x \geqslant 1\).

1. Find an expression for \(\mathrm { f } ^ { \prime } ( x )\).
2. Determine, with a reason, whether \(f\) is an increasing function, a decreasing function or neither.
3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\), and state the domain and range of \(\mathrm { f } ^ { - 1 }\).

5 A function f is such that \(\mathrm { f } ( x ) = \frac { 15 } { 2 x + 3 }\) for \(0 \leqslant x \leqslant 6\).

1. Find an expression for \(\mathrm { f } ^ { \prime } ( x )\) and use your result to explain why f has an inverse.
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\), and state the domain and range of \(\mathrm { f } ^ { - 1 }\).

6 \includegraphics[max width=\textwidth, alt={}, center]{8da9e73a-3126-471b-b904-25e3c156f6bf-2_519_670_1640_735} The diagram shows the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\), where \(\mathrm { f } ^ { - 1 }\) is defined by \(\mathrm { f } ^ { - 1 } ( x ) = \frac { 1 - 5 x } { 2 x }\) for \(0 < x \leqslant 2\).

1. Find an expression for \(\mathrm { f } ( x )\) and state the domain of f .
2. The function g is defined by \(\mathrm { g } ( x ) = \frac { 1 } { x }\) for \(x \geqslant 1\). Find an expression for \(\mathrm { f } ^ { - 1 } \mathrm {~g} ( x )\), giving your answer in the form \(a x + b\), where \(a\) and \(b\) are constants to be found.

11 The function f is defined by \(\mathrm { f } : x \mapsto 4 \sin x - 1\) for \(- \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi\).

1. State the range of f .
2. Find the coordinates of the points at which the curve \(y = \mathrm { f } ( x )\) intersects the coordinate axes.
3. Sketch the graph of \(y = \mathrm { f } ( x )\).
4. Obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\), stating both the domain and range of \(\mathrm { f } ^ { - 1 }\). {www.cie.org.uk} after the live examination series. }

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

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

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

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 \(\operatorname { ff } ( x ) = \frac { 185 } { 16 }\).

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

3. $$f ( x ) = 3 - \frac { x - 2 } { x + 1 } + \frac { 5 x + 26 } { 2 x ^ { 2 } - 3 x - 5 } \quad x > 4$$

1. Show that $$\mathrm { f } ( x ) = \frac { a x + b } { c x + d } \quad x > 4$$ where \(a , b , c\) and \(d\) are integers to be found.
2. Hence find \(\mathrm { f } ^ { - 1 } ( x )\)
3. Find the domain of \(\mathrm { f } ^ { - 1 }\)

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.