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

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

10 The functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto 3 x + 2 , & x \in \mathbb { R } , \\ \mathrm {~g} : x \mapsto \frac { 6 } { 2 x + 3 } , & x \in \mathbb { R } , x \neq - 1.5 . \end{array}$$

1. Find the value of \(x\) for which \(\operatorname { fg } ( x ) = 3\).
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 two graphs.
3. Express each of \(\mathrm { f } ^ { - 1 } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\) in terms of \(x\), and solve the equation \(\mathrm { f } ^ { - 1 } ( x ) = \mathrm { g } ^ { - 1 } ( x )\).

5 The function f is defined by \(\mathrm { f } : x \mapsto a x + b\), for \(x \in \mathbb { R }\), where \(a\) and \(b\) are constants. It is given that \(f ( 2 ) = 1\) and \(f ( 5 ) = 7\).

1. Find the values of \(a\) and \(b\).
2. Solve the equation \(\operatorname { ff } ( x ) = 0\).

11 The equation of a curve is \(y = 8 x - x ^ { 2 }\).

1. Express \(8 x - x ^ { 2 }\) in the form \(a - ( x + b ) ^ { 2 }\), stating the numerical values of \(a\) and \(b\).
2. Hence, or otherwise, find the coordinates of the stationary point of the curve.
3. Find the set of values of \(x\) for which \(y \geqslant - 20\). The function g is defined by \(\mathrm { g } : x \mapsto 8 x - x ^ { 2 }\), for \(x \geqslant 4\).
4. State the domain and range of \(\mathrm { g } ^ { - 1 }\).
5. Find an expression, in terms of \(x\), for \(\mathrm { g } ^ { - 1 } ( x )\).

10 The functions \(f\) and \(g\) are defined as follows: $$\begin{array} { l l } \mathrm { f } : x \mapsto x ^ { 2 } - 2 x , & x \in \mathbb { R } , \\ \mathrm {~g} : x \mapsto 2 x + 3 , & x \in \mathbb { R } . \end{array}$$

1. Find the set of values of \(x\) for which \(\mathrm { f } ( x ) > 15\).
2. Find the range of f and state, with a reason, whether f has an inverse.
3. Show that the equation \(\operatorname { gf } ( x ) = 0\) has no real solutions.
4. Sketch, in a single diagram, the graphs of \(y = \mathrm { g } ( x )\) and \(y = \mathrm { g } ^ { - 1 } ( x )\), making clear the relationship between the graphs.

7 A function f is defined by f : \(x \mapsto 3 - 2 \sin x\), for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

1. Find the range of f .
2. Sketch the graph of \(y = \mathrm { f } ( x )\). A function g is defined by \(\mathrm { g } : x \mapsto 3 - 2 \sin x\), for \(0 ^ { \circ } \leqslant x \leqslant A ^ { \circ }\), where \(A\) is a constant.
3. State the largest value of \(A\) for which g has an inverse.
4. When \(A\) has this value, obtain an expression, in terms of \(x\), for \(\mathrm { g } ^ { - 1 } ( x )\).

11 Functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto k - x & \text { for } x \in \mathbb { R } , \text { where } k \text { is a constant, } \\ \mathrm { g } : x \mapsto \frac { 9 } { x + 2 } & \text { for } x \in \mathbb { R } , x \neq - 2 . \end{array}$$

1. Find the values of \(k\) for which the equation \(\mathrm { f } ( x ) = \mathrm { g } ( x )\) has two equal roots and solve the equation \(\mathrm { f } ( x ) = \mathrm { g } ( x )\) in these cases.
2. Solve the equation \(\operatorname { fg } ( x ) = 5\) when \(k = 6\).
3. Express \(\mathrm { g } ^ { - 1 } ( x )\) in terms of \(x\).

11 \includegraphics[max width=\textwidth, alt={}, center]{b24ed4c7-ab07-45f4-adf2-027734c36b62-4_862_892_932_628} The diagram shows the graph of \(y = \mathrm { f } ( x )\), where \(\mathrm { f } : x \mapsto \frac { 6 } { 2 x + 3 }\) for \(x \geqslant 0\).

1. Find an expression, in terms of \(x\), for \(\mathrm { f } ^ { \prime } ( x )\) and explain how your answer shows that f is a decreasing function.
2. Find an expression, in terms of \(x\), for \(\mathrm { f } ^ { - 1 } ( x )\) and find the domain of \(\mathrm { f } ^ { - 1 }\).
3. Copy the diagram and, on your copy, sketch the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\), making clear the relationship between the graphs. The function g is defined by \(\mathrm { g } : x \mapsto \frac { 1 } { 2 } x\) for \(x \geqslant 0\).
4. Solve the equation \(\operatorname { fg } ( x ) = \frac { 3 } { 2 }\).

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

8 Functions f and g are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto 4 x - 2 k & \text { for } x \in \mathbb { R } , \text { where } k \text { is a constant, } \\ \mathrm { g } : x \mapsto \frac { 9 } { 2 - x } & \text { for } x \in \mathbb { R } , x \neq 2 . \end{array}$$

1. Find the values of \(k\) for which the equation \(\mathrm { fg } ( x ) = x\) has two equal roots.
2. Determine the roots of the equation \(\operatorname { fg } ( x ) = x\) for the values of \(k\) found in part (i).

10 The function f is defined by \(\mathrm { f } : x \mapsto 2 x ^ { 2 } - 12 x + 13\) for \(0 \leqslant x \leqslant A\), where \(A\) is a constant.

1. Express \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
2. State the value of \(A\) for which the graph of \(y = \mathrm { f } ( x )\) has a line of symmetry.
3. When \(A\) has this value, find the range of f . The function g is defined by \(\mathrm { g } : x \mapsto 2 x ^ { 2 } - 12 x + 13\) for \(x \geqslant 4\).
4. Explain why \(g\) has an inverse.
5. Obtain an expression, in terms of \(x\), for \(\mathrm { g } ^ { - 1 } ( x )\).

10 The function \(\mathrm { f } : x \mapsto 2 x ^ { 2 } - 8 x + 14\) is defined for \(x \in \mathbb { R }\).

1. Find the values of the constant \(k\) for which the line \(y + k x = 12\) is a tangent to the curve \(y = \mathrm { f } ( x )\).
2. Express \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
3. Find the range of f . The function \(\mathrm { g } : x \mapsto 2 x ^ { 2 } - 8 x + 14\) is defined for \(x \geqslant A\).
4. Find the smallest value of \(A\) for which g has an inverse.
5. For this value of \(A\), find an expression for \(\mathrm { g } ^ { - 1 } ( x )\) in terms of \(x\).

11 Functions f and g are defined for \(x \in \mathbb { R }\) by $$\begin{aligned} & \mathrm { f } : x \mapsto 2 x + 1 \\ & \mathrm {~g} : x \mapsto x ^ { 2 } - 2 \end{aligned}$$

1. Find and simplify expressions for \(\mathrm { fg } ( x )\) and \(\mathrm { gf } ( x )\).
2. Hence find the value of \(a\) for which \(\mathrm { fg } ( a ) = \mathrm { gf } ( a )\).
3. Find the value of \(b ( b \neq a )\) for which \(\mathrm { g } ( b ) = b\).
4. Find and simplify an expression for \(\mathrm { f } ^ { - 1 } \mathrm {~g} ( x )\). The function h is defined by $$\mathrm { h } : x \mapsto x ^ { 2 } - 2 , \quad \text { for } x \leqslant 0$$
5. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).

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 \(x\).

8 The function \(\mathrm { f } : x \mapsto x ^ { 2 } - 4 x + k\) is defined for the domain \(x \geqslant p\), where \(k\) and \(p\) are constants.

1. Express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b + k\), where \(a\) and \(b\) are constants.
2. State the range of f in terms of \(k\).
3. State the smallest value of \(p\) for which f is one-one.
4. For the value of \(p\) found in part (iii), find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain \(\mathrm { f } ^ { - 1 }\), giving your answers in terms of \(k\).

8

1. Express \(2 x ^ { 2 } - 12 x + 13\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
2. The function f is defined by \(\mathrm { f } ( x ) = 2 x ^ { 2 } - 12 x + 13\) for \(x \geqslant k\), where \(k\) is a constant. It is given that f is a one-one function. State the smallest possible value of \(k\). The value of \(k\) is now given to be 7 .
3. Find the range of f .
4. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).

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

9

1. In an arithmetic progression, the sum, \(S _ { n }\), of the first \(n\) terms is given by \(S _ { n } = 2 n ^ { 2 } + 8 n\). Find the first term and the common difference of the progression.
2. The first 2 terms of a geometric progression are 64 and 48 respectively. The first 3 terms of the geometric progression are also the 1st term, the 9th term and the \(n\)th term respectively of an arithmetic progression. Find the value of \(n\).

10 The function f is defined by \(\mathrm { f } : x \mapsto 2 x + k , x \in \mathbb { R }\), where \(k\) is a constant.

1. In the case where \(k = 3\), solve the equation \(\mathrm { ff } ( x ) = 25\). The function g is defined by \(\mathrm { g } : x \mapsto x ^ { 2 } - 6 x + 8 , x \in \mathbb { R }\).
2. Find the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = \mathrm { g } ( x )\) has no real solutions. The function \(h\) is defined by \(h : x \mapsto x ^ { 2 } - 6 x + 8 , x > 3\).
3. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).

10 \includegraphics[max width=\textwidth, alt={}, center]{62f7f1e2-a8e7-4574-a432-8e9b20b54d7a-4_819_812_255_662} The diagram shows the function f defined for \(- 1 \leqslant x \leqslant 4\), where $$f ( x ) = \begin{cases} 3 x - 2 & \text { for } - 1 \leqslant x \leqslant 1 \\ \frac { 4 } { 5 - x } & \text { for } 1 < x \leqslant 4 \end{cases}$$

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 the function \(\mathrm { f } ^ { - 1 }\), giving also the set of values for which each expression is valid.

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

1. Solve the equation \(\mathrm { ff } ( x ) = 11\).
2. Find the range of g .
3. Find the set of values of \(x\) for which \(\mathrm { g } ( x ) > 12\).
4. Find the value of the constant \(p\) for which the equation \(\mathrm { gf } ( x ) = p\) has two equal roots. Function h is defined by \(\mathrm { h } : x \mapsto x ^ { 2 } + 4 x\) for \(x \geqslant k\), and it is given that h has an inverse.
5. State the smallest possible value of \(k\).
6. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).

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. }

1 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 10 - 3 x , \quad x \in \mathbb { R } , \\ & \mathrm {~g} : x \mapsto \frac { 10 } { 3 - 2 x } , \quad x \in \mathbb { R } , x \neq \frac { 3 } { 2 } \end{aligned}$$ Solve the equation \(\mathrm { ff } ( x ) = \mathrm { gf } ( 2 )\).