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

5 The functions \(f\) and \(g\) are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = 2 - x ^ { 4 } & \text { for all real values of } x \\ \mathrm {~g} ( x ) = \frac { 1 } { x - 4 } & \text { for real values of } x , x \neq 4 \end{array}$$

1. State the range of f .
2. Explain why the function f does not have an inverse.
3. 1. Write down an expression for fg(x).
   2. Solve the equation \(\operatorname { fg } ( x ) = - 14\).

6 The functions \(f\) and \(g\) are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } - 3 , & \text { for all real values of } x \\ \mathrm {~g} ( x ) = \frac { 1 } { 3 x + 4 } , & \text { for real values of } x , x \neq - \frac { 4 } { 3 } \end{array}$$

1. Find the range of \(f\).
2. The inverse of f is \(\mathrm { f } ^ { - 1 }\).
   1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
   2. Solve the equation \(\mathrm { f } ^ { - 1 } ( x ) = 0\).
3. 1. Find an expression for \(\operatorname { gf } ( x )\).
   2. Solve the equation \(\mathrm { gf } ( x ) = 1\), giving your answer in an exact form.

5 The functions \(f\) and \(g\) are defined with their respective domains by $$\begin{aligned} & \mathrm { f } ( x ) = \sqrt { x - 2 } \text { for } x \geqslant 2 \\ & \mathrm {~g} ( x ) = \frac { 1 } { x } \quad \text { for real values of } x , x \neq 0 \end{aligned}$$

1. State the range of f .
2. 1. Find fg(x).
   2. Solve the equation \(\operatorname { fg } ( x ) = 1\).
3. The inverse of f is \(\mathrm { f } ^ { - 1 }\). Find \(\mathrm { f } ^ { - 1 } ( x )\).

4 The functions f and g are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = 5 - \mathrm { e } ^ { 3 x } , & \text { for all real values of } x \\ \mathrm {~g} ( x ) = \frac { 1 } { 2 x - 3 } , & \text { for } x \neq 1.5 \end{array}$$

1. Find the range of f.
2. The inverse of f is \(\mathrm { f } ^ { - 1 }\).
   1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
   2. Solve the equation \(\mathrm { f } ^ { - 1 } ( x ) = 0\).
3. Find an expression for \(\operatorname { gg } ( x )\), giving your answer in the form \(\frac { a x + b } { c x + d }\), where \(a , b , c\) and \(d\) are integers.
   [0pt] [3 marks]

9 The function f is defined by $$f ( x ) = \frac { x ^ { 2 } + 2 x + 2 } { x ^ { 2 } }$$

1. Write down the equations of the two asymptotes to the curve \(y = \mathrm { f } ( x )\).
2. By considering the expression \(x ^ { 2 } + 2 x + 2\) :
   1. show that the graph of \(y = \mathrm { f } ( x )\) does not intersect the \(x\)-axis;
   2. find the non-real roots of the equation \(\mathrm { f } ( x ) = 0\).
3. 1. Show that, if the equation \(\mathrm { f } ( x ) = k\) has two equal roots, then $$4 - 8 ( 1 - k ) = 0$$
   2. Deduce that the graph of \(y = \mathrm { f } ( x )\) has exactly one stationary point and find its coordinates.

4.A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where \(x \in \mathbb { R }\) and f is a one-one function.

1. Describe a single transformation that transforms \(C\) to the curve with equation \(y = - \mathrm { f } ( - x )\) . The curve \(C\) is reflected in the line with equation \(y = - x\) to give the curve \(V\) . The equation of \(V\) is \(y = \mathrm { g } ( x )\) .
2. Explain why \(\mathrm { g } ^ { - 1 } ( x ) = - \mathrm { f } ( - x )\) . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2a7c2530-a93c-4a26-bc37-c20c0f40c8f2-3_793_979_819_633} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \frac { 3 ( x - 1 ) } { x - 2 } \quad x \in \mathbb { R } , x \neq 2$$ The curve has asymptotes with equations \(x = p\) and \(y = q\) and \(C\) crosses the \(x\)-axis at the point \(A\) and the \(y\)-axis at the point \(B\) .
3. Write down the value of \(p\) and the value of \(q\) .
4. Write down the coordinates of the point \(A\) and the coordinates of the point \(B\) . Given the definition of \(\mathrm { g } ( x )\) in part(b),
5. find the function g .
6. Solve \(\mathrm { g } ^ { - 1 } \mathrm { f } ( x ) = x\)

11 For all real values of \(x\), the functions f and g are defined by \(\mathrm { f } ( x ) = x ^ { 2 } + 8 a x + 4 a ^ { 2 }\) and \(g ( x ) = 6 x - 2 a\), where \(a\) is a positive constant.

1. Find \(\mathrm { fg } ( x )\). Determine the range of \(\mathrm { fg } ( x )\) in terms of \(a\).
2. If \(f g ( 2 ) = 144\), find the value of \(a\).
3. Determine whether the function fg has an inverse.

6

1. The diagrams show five different graphs. In each case the whole of the graph is shown. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cb83836f-753f-4b3a-99e8-a18aff0f49ff-06_376_382_310_306} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cb83836f-753f-4b3a-99e8-a18aff0f49ff-06_376_378_310_842} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cb83836f-753f-4b3a-99e8-a18aff0f49ff-06_378_378_310_1379} \captionsetup{labelformat=empty} \caption{Fig. 1.3}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cb83836f-753f-4b3a-99e8-a18aff0f49ff-06_378_382_872_306} \captionsetup{labelformat=empty} \caption{Fig. 1.4}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cb83836f-753f-4b3a-99e8-a18aff0f49ff-06_378_378_872_845} \captionsetup{labelformat=empty} \caption{Fig. 1.5}
   \end{figure} Place ticks in the boxes in the table in the Printed Answer Booklet to indicate, for each graph, whether it represents a one-one function, a many-one function, a function that is its own inverse or it does not represent a function. There may be more than one tick in some rows or columns of the table.
2. A function f is defined by \(\mathrm { f } ( x ) = \frac { 1 } { x }\) for the domain \(\{ x : 0 < x \leqslant 2 \}\). State the range of f , giving your answer in set notation.

4 The function f is defined by \(\mathrm { f } ( x ) = \mathrm { e } ^ { x - 4 } , x \in \mathbb { R }\) Find \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.

13 The function f is defined by $$\mathrm { f } ( x ) = \frac { 2 x + 3 } { x - 2 } \quad x \in \mathbb { R } , x \neq 2$$ 13

1. 1. Find f-1
      13
      1. (ii) Write down an expression for \(\mathrm { ff } ( x )\).
         13
   2. The function g is defined by $$g ( x ) = \frac { 2 x ^ { 2 } - 5 x } { 2 } \quad x \in \mathbb { R } , 0 \leq x \leq 4$$ 13
   3. 1. Find the range of g .
         13
   4. (ii) Determine whether g has an inverse.
      Fully justify your answer.
   5. Show that $$g f ( x ) = \frac { 48 + 29 x - 2 x ^ { 2 } } { 2 x ^ { 2 } - 8 x + 8 }$$ 13
   6. It can be shown that fg is given by $$f g ( x ) = \frac { 4 x ^ { 2 } - 10 x + 6 } { 2 x ^ { 2 } - 5 x - 4 }$$ with domain \(\{ x \in \mathbb { R } : 0 \leq x \leq 4 , x \neq a \}\) Find the value of \(a\).
      Fully justify your answer.

7 The functions f and g are defined by $$\begin{aligned} & \mathrm { f } ( x ) = \sqrt { 10 - 2 x } \text { for } \quad x \leq 5 \\ & \mathrm {~g} ( x ) = \frac { 1 } { x } \quad \text { for } \quad x \neq 0 \end{aligned}$$ The function \(h\) has maximum possible domain and is defined by $$\mathrm { h } ( x ) = \operatorname { gf } ( x )$$ 7

1. Find an expression for \(\mathrm { h } ( x )\) 7
2. Find the domain of h
   7
3. Show that \(\mathrm { h } ^ { - 1 } ( x ) = 5 - \frac { 1 } { 2 x ^ { 2 } }\) \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-11_2488_1716_219_153}

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

1. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
2. Sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) on the same diagram.
3. Find the set of values of \(x\) for which \(\mathrm { f } ( x ) > \mathrm { f } ^ { - 1 } ( x )\).

13

1. Write down the equations of the asymptotes of curve \(C _ { 1 }\) 13 A curve \(C _ { 1 }\) has equation 13
2. On the axes below, sketch the graph of curve \(C _ { 1 }\) Indicate the values of the intercepts of the curve with the axes. \includegraphics[max width=\textwidth, alt={}, center]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-20_885_898_1192_571} 13
3. Hence, or otherwise, solve the inequality $$\frac { 2 x + 7 } { 3 x + 5 } \geq 0$$ 13
4. Curve \(C _ { 2 }\) is a reflection of curve \(C _ { 1 }\) in the line \(y = - x\) Find an equation for curve \(C _ { 2 }\) in the form \(y = \mathrm { f } ( x )\)

9 The function f is defined by $$f ( x ) = \frac { x ( x + 3 ) } { x + 4 } \quad ( x \in \mathbb { R } , x \neq - 4 )$$ 9

1. Find the interval ( \(a , b\) ) in which \(\mathrm { f } ( x )\) does not take any values.
   Fully justify your answer.
   9
2. Find the coordinates of the two stationary points of the graph of \(y = \mathrm { f } ( x )\) 9
3. Show that the graph of \(y = \mathrm { f } ( x )\) has an oblique asymptote and find its equation.
   \section*{Question 9 continues on the next page} 9
4. Sketch the graph of \(y = \mathrm { f } ( x )\) on the axes below.
   [0pt] [4 marks] \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-16_1100_1100_406_470} \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-17_2493_1732_214_139}
   1. Fird \(\begin{aligned} & \text { Do not write } \\ & \text { outside the } \end{aligned}\)

1. The functions f and g are defined by

$$\begin{array} { l l } \mathrm { f } ( x ) = 6 - \frac { 21 } { 2 x + 3 } & x \geqslant 0 \\ \mathrm {~g} ( x ) = x ^ { 2 } + 5 & x \in \mathbb { R } \end{array}$$

1. Find \(\mathrm { gf } ( 2 )\)
2. Find \(f ^ { - 1 }\)
3. Solve the equation $$\operatorname { gg } ( x ) = 126$$

5 Let \(\mathrm { f } ( x ) = x ^ { 2 }\) and \(\mathrm { g } ( x ) = 7 x - 2\) for all real values of \(x\).

1. Give a reason why f has no inverse function.
2. Write down an expression for \(\mathrm { gf } ( x )\).
3. Find \(\mathrm { g } ^ { - 1 } ( x )\).

11 The functions f and g are defined by \(\mathrm { f } ( x ) = \frac { 1 } { 2 + x } + 5 , x > - 2\) and \(\mathrm { g } ( x ) = | x | , x \in \mathbb { R }\).

1. Given that the range of f is of the form \(\mathrm { f } ( x ) > a\), find \(a\).
2. Find an expression for \(\mathrm { f } ^ { - 1 }\), stating its domain and range.
3. Show that \(\mathrm { gf } ( x ) = \mathrm { f } ( x )\).
4. Find an expression for \(\mathrm { fg } ( x )\). Determine whether fg has an inverse.

6 The functions f and g are given by \(\mathrm { f } ( x ) = \frac { 3 } { x - 1 }\) for all \(x \neq 1\) and \(\mathrm { g } ( x ) = x + 2\) for all real \(x\).

1. Find gf, stating its domain and range.
2. Find \(( \mathrm { gf } ) ^ { - 1 }\), stating any values of \(x\) for which \(( \mathrm { gf } ) ^ { - 1 }\) is not defined.

7 The functions f and g are defined for all real numbers by $$\mathrm { f } ( x ) = x ^ { 2 } + 2 \quad \text { and } \quad \mathrm { g } ( x ) = 4 x + 3$$

1. State the range of each of the functions f and g .
2. Find the values of \(x\) for which \(\mathrm { fg } ( x ) = \mathrm { gf } ( x )\).
3. The function h , given by \(\mathrm { h } ( x ) = x ^ { 2 } + 2\), has the same range as f but is such that \(\mathrm { h } ^ { - 1 } ( x )\) exists. State a possible domain for h and find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).

3 Let \(\mathrm { f } ( x ) = x ^ { 2 }\) and \(\mathrm { g } ( x ) = 7 x - 2\) for all real values of \(x\).

1. Give a reason why f has no inverse function.
2. Write down an expression for \(\operatorname { gf } ( x )\).
3. Find \(\mathrm { g } ^ { - 1 } ( x )\).
4. Explain the relationship between the graph of \(y = \mathrm { g } ( x )\) and \(y = \mathrm { g } ^ { - 1 } ( x )\).

2 It is given that \(\mathrm { f } ( x ) = 4 + 3 \sqrt { x }\), where \(x \geqslant 0\).

1. State the range of f .
2. State the value of \(\mathrm { ff } ( 16 )\).
3. Find \(\mathrm { f } ^ { - 1 } ( x )\).
4. On the same axes, sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) and state how the graphs are related.

3 Let \(\mathrm { f } ( x ) = x ^ { 2 }\) and \(\mathrm { g } ( x ) = 7 x - 2\) for all real values of \(x\).

1. Give a reason why f has no inverse function.
2. Write down an expression for \(\operatorname { gf } ( x )\).
3. Find \(\mathrm { g } ^ { - 1 } ( x )\).
4. Explain the relationship between the graph of \(y = \mathrm { g } ( x )\) and \(y = \mathrm { g } ^ { - 1 } ( x )\).

2 The function f is defined by \(\mathrm { f } ( x ) = \frac { 2 x + 1 } { x - 3 }\) for all real \(x , x \neq 3\). Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).

a) Differentiate each of the following functions with respect to \(x\). i) \(x ^ { 5 } \ln x\) ii) \(\frac { \mathrm { e } ^ { 3 x } } { x ^ { 3 } - 1 }\) iii) \(( \tan x + 7 x ) ^ { \frac { 1 } { 2 } }\) b) A function is defined implicitly by $$3 y + 4 x y ^ { 2 } - 5 x ^ { 3 } = 8$$ Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
□
1
The function \(f ( x )\) is defined by $$f ( x ) = \frac { \sqrt { x ^ { 2 } - 1 } } { x }$$ with domain \(x \geqslant 1\).
a) Find an expression for \(f ^ { - 1 } ( x )\). State the domain for \(f ^ { - 1 }\) and sketch both \(f ( x )\) and \(f ^ { - 1 } ( x )\) on the same diagram.
b) Explain why the function \(f f ( x )\) cannot be formed.
□
1
A chord \(A B\) subtends an angle \(\theta\) radians at the centre of a circle. The chord divides the circle into two segments whose areas are in the ratio \(1 : 2\). \includegraphics[max width=\textwidth, alt={}, center]{966abb82-ade0-4ca8-87a4-26e806d5add7-5_572_576_1197_749}
a) Show that \(\sin \theta = \theta - \frac { 2 \pi } { 3 }\).
b) i) Show that \(\theta\) lies between \(2 \cdot 6\) and \(2 \cdot 7\).
ii) Starting with \(\theta _ { 0 } = 2 \cdot 6\), use the Newton-Raphson Method to find the value of \(\theta\) correct to three decimal places. \section*{TURN OVER} Wildflowers grow on the grass verge by the side of a motorway. The area populated by wildflowers at time \(t\) years is \(A \mathrm {~m} ^ { 2 }\). The rate of increase of \(A\) is directly proportional to \(A\).
a) Write down a differential equation that is satisfied by \(A\).
b) At time \(t = 0\), the area populated by wildflowers is \(0.2 \mathrm {~m} ^ { 2 }\). One year later, the area has increased to \(1.48 \mathrm {~m} ^ { 2 }\). Find an expression for \(A\) in terms of \(t\) in the form \(p q ^ { t }\), where \(p\) and \(q\) are rational numbers to be determined.

The diagram below shows a sketch of the graph of \(y = f ( x )\), where $$f ( x ) = 2 x ^ { 2 } + 12 x + 10 .$$ The graph intersects the \(x\)-axis at the points \(( p , 0 ) , ( q , 0 )\) and the \(y\)-axis at the point \(( 0,10 )\). \includegraphics[max width=\textwidth, alt={}, center]{72bb1603-edbd-4e2e-bf2b-f33bb667e61b-5_1004_1171_648_440}
a) Write down the value of \(f f ( p )\).
b) Determine the values of \(p\) and \(q\).
c) Express \(f ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b , c\) are constants whose values are to be found. Write down the coordinates of the minimum point.
d) Explain why \(f ^ { - 1 } ( x )\) does not exist.
e) The function \(g ( x )\) is defined as $$g ( x ) = f ( x ) \quad \text { for } \quad - 3 \leqslant x < \infty .$$ i) Find an expression for \(g ^ { - 1 } ( x )\).
ii) Sketch the graph of \(y = g ^ { - 1 } ( x )\), indicating the coordinates of the points where the graph intersects the \(x\)-axis and the \(y\)-axis.