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

1. The functions \(f\) and \(g\) are defined by

$$\begin{aligned} & \mathrm { f } : x \rightarrow 7 x - 1 , \quad x \in \mathbb { R } \\ & \mathrm {~g} : x \rightarrow \frac { 4 } { x - 2 } , \quad x \neq 2 , x \in \mathbb { R } \end{aligned}$$

1. Solve the equation \(\operatorname { fg } ( x ) = x\)
2. Hence, or otherwise, find the largest value of \(a\) such that \(\mathrm { g } ( a ) = \mathrm { f } ^ { - 1 } ( a )\)

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the graph of \(y = \mathrm { g } ( x )\), where $$g ( x ) = 3 + \sqrt { x + 2 } , \quad x \geqslant - 2$$

1. State the range of g .
2. Find \(\mathrm { g } ^ { - 1 } ( x )\) and state its domain.
3. Find the exact value of \(x\) for which $$\mathrm { g } ( x ) = x$$
4. Hence state the value of \(a\) for which $$\mathrm { g } ( a ) = \mathrm { g } ^ { - 1 } ( a )$$

1. The function f is defined by

$$\mathrm { f } ( x ) = \frac { 6 } { 2 x + 5 } + \frac { 2 } { 2 x - 5 } + \frac { 60 } { 4 x ^ { 2 } - 25 } , \quad x > 4$$

1. Show that \(\mathrm { f } ( x ) = \frac { A } { B x + C }\) where \(A , B\) and \(C\) are constants to be found.
2. Find \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.
   

2. The function \(f\) is defined by $$\mathrm { f } : x \mapsto 2 x , \quad x \in \mathbb { R }$$

1. Find \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\). The function g is defined by $$\mathrm { g } : x \mapsto 3 x ^ { 2 } + 2 , \quad x \in \mathbb { R }$$
2. Find \(\mathrm { gf } ^ { - 1 } ( x )\).
3. State the range of \(\mathrm { gf } ^ { - 1 } ( x )\).

1. The function f is defined by

$$\mathrm { f } : x \mapsto | x - 2 | - 3 , x \in \mathbb { R }$$

1. Solve the equation \(\mathrm { f } ( x ) = 1\). The function g is defined by $$\mathrm { g } : x \mapsto x ^ { 2 } - 4 x + 11 , x \geq 0$$
2. Find the range of g .
3. Find \(g f ( - 1 )\).
   

\begin{figure}[h] \end{figure} Figure 1 shows part of the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\), where f is an increasing function of \(x\). The curve passes through the points \(P ( 0 , - 2 )\) and \(Q ( 3,0 )\) as shown. In separate diagrams, sketch the curve with equation

1. \(y = | f ( x ) |\),
2. \(y = \mathrm { f } ^ { - 1 } ( x )\),
3. \(y = \frac { 1 } { 2 } \mathrm { f } ( 3 x )\). Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.

2 The functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined for all real numbers \(x\) by $$\mathrm { f } ( x ) = x ^ { 2 } , \quad \mathrm {~g} ( x ) = x - 2$$

1. Find the composite functions \(\mathrm { fg } ( x )\) and \(\mathrm { gf } ( x )\).
2. Sketch the curves \(y = \mathrm { f } ( x ) , y = \mathrm { fg } ( x )\) and \(y = \mathrm { gf } ( x )\), indicating clearly which is which.

8 Fig. 8 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = 1 + \sin 2 x\) for \(- \frac { 1 } { 4 } \pi \leqslant x \leqslant \frac { 1 } { 4 } \pi\).

1. State a sequence of two transformations that would map part of the curve \(y = \sin x\) onto the curve \(y = \mathrm { f } ( x )\).
2. Find the area of the region enclosed by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis and the line \(x = \frac { 1 } { 4 } \pi\).
3. Find the gradient of the curve \(y = \mathrm { f } ( x )\) at the point \(( 0,1 )\). Hence write down the gradient of the curve \(y = \mathrm { f } ^ { - 1 } ( x )\) at the point \(( 1,0 )\).
4. State the domain of \(\mathrm { f } ^ { - 1 } ( x )\). Add a sketch of \(y = \mathrm { f } ^ { - 1 } ( x )\) to a copy of Fig. 8.
5. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).

9 The function \(\mathrm { f } ( x ) = \ln \left( 1 + x ^ { 2 } \right)\) has domain \(- 3 \leqslant x \leqslant 3\).
Fig. 9 shows the graph of \(y = \mathrm { f } ( x )\). \begin{figure}[h] \end{figure}

1. Show algebraically that the function is even. State how this property relates to the shape of the curve.
2. Find the gradient of the curve at the point \(\mathrm { P } ( 2 , \ln 5 )\).
3. Explain why the function does not have an inverse for the domain \(- 3 \leqslant x \leqslant 3\). The domain of \(\mathrm { f } ( x )\) is now restricted to \(0 \leqslant x \leqslant 3\). The inverse of \(\mathrm { f } ( x )\) is the function \(\mathrm { g } ( x )\).
4. Sketch the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\) on the same axes. State the domain of the function \(\mathrm { g } ( x )\). Show that \(\mathrm { g } ( x ) = \sqrt { \mathrm { e } ^ { x } - 1 }\).
5. Differentiate \(\mathrm { g } ( x )\). Hence verify that \(\mathrm { g } ^ { \prime } ( \ln 5 ) = 1 \frac { 1 } { 4 }\). Explain the connection between this result and your answer to part (ii).

6. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \rightarrow 3 x - 4 , \quad x \in \mathbb { R } , \\ & \mathrm {~g} : x \rightarrow \frac { 2 } { x + 3 } , \quad x \in \mathbb { R } , \quad x \neq - 3 \end{aligned}$$

1. Evaluate fg(1).
2. Solve the equation \(\operatorname { gf } ( x ) = 6\).
3. Find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).

9. \(\quad f ( x ) = 3 - e ^ { 2 x } , \quad x \in \mathbb { R }\).

1. State the range of f .
2. Find the exact value of \(\mathrm { ff } ( 0 )\).
3. Define the inverse function \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain. Given that \(\alpha\) is the solution of the equation \(\mathrm { f } ( x ) = \mathrm { f } ^ { - 1 } ( x )\),
4. explain why \(\alpha\) satisfies the equation $$x = \mathrm { f } ^ { - 1 } ( x )$$
5. use the iterative formula $$x _ { n + 1 } = \mathrm { f } ^ { - 1 } \left( x _ { n } \right)$$ with \(x _ { 0 } = 0.5\) to find \(\alpha\) correct to 3 significant figures.

5. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } ( x ) \equiv x ^ { 2 } - 3 x + 7 , \quad x \in \mathbb { R } , \\ & \mathrm {~g} ( x ) \equiv 2 x - 1 , \quad x \in \mathbb { R } . \end{aligned}$$

1. Find the range of f .
2. Evaluate \(g f ( - 1 )\).
3. Solve the equation $$\operatorname { fg } ( x ) = 17$$

7. \includegraphics[max width=\textwidth, alt={}, center]{039ebdba-4ad5-4974-9345-d66712fa0a08-3_401_712_228_479} The diagram shows the graph of \(y = \mathrm { f } ( x )\) which meets the coordinate axes at the points ( \(a , 0\) ) and ( \(0 , b\) ), where \(a\) and \(b\) are constants.

1. Showing, in terms of \(a\) and \(b\), the coordinates of any points of intersection with the axes, sketch on separate diagrams the graphs of
   1. \(y = \mathrm { f } ^ { - 1 } ( x )\),
   2. \(y = 2 \mathrm { f } ( 3 x )\). Given that $$\mathrm { f } ( x ) = 2 - \sqrt { x + 9 } , \quad x \in \mathbb { R } , \quad x \geq - 9$$
2. find the values of \(a\) and \(b\),
3. find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.

7. The function \(f\) is defined by $$\mathrm { f } : x \rightarrow 3 \mathrm { e } ^ { x - 1 } , \quad x \in \mathbb { R }$$

1. State the range of f .
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain. The function \(g\) is defined by $$g : x \rightarrow 5 x - 2 , \quad x \in \mathbb { R }$$ Find, in terms of e,
3. the value of \(\mathrm { gf } ( \ln 2 )\),
4. the solution of the equation $$\mathrm { f } ^ { - 1 } \mathrm {~g} ( x ) = 4$$

5. The function \(f\) is defined by $$\mathrm { f } ( x ) \equiv 4 - \ln 3 x , \quad x \in \mathbb { R } , \quad x > 0$$

1. Solve the equation \(\mathrm { f } ( x ) = 0\).
2. Sketch the curve \(y = \mathrm { f } ( x )\). The function g is defined by $$\mathrm { g } ( x ) \equiv \mathrm { e } ^ { 2 - x } , \quad x \in \mathbb { R }$$
3. Show that $$\operatorname { fg } ( x ) = x + a - \ln b$$ where \(a\) and \(b\) are integers to be found.

8. $$\mathrm { f } ( x ) \equiv 2 x ^ { 2 } + 4 x + 2 , \quad x \in \mathbb { R } , \quad x \geq - 1$$

1. Express \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\).
2. Describe fully two transformations that would map the graph of \(y = x ^ { 2 } , x \geq 0\) onto the graph of \(y = \mathrm { f } ( x )\).
3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.
4. Sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) on the same diagram and state the relationship between them.

2. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & f : x \rightarrow 2 - x ^ { 2 } , \quad x \in \mathbb { R } , \\ & g : x \rightarrow \frac { 3 x } { 2 x - 1 } , \quad x \in \mathbb { R } , \quad x \neq \frac { 1 } { 2 } . \end{aligned}$$

1. Evaluate fg(2).
2. Solve the equation \(\operatorname { gf } ( x ) = \frac { 1 } { 2 }\).

9. \includegraphics[max width=\textwidth, alt={}, center]{5e6a37a1-c51f-4637-aaae-48da6ab3eca0-3_727_1022_244_342} The diagram shows the curve with equation \(y = \mathrm { f } ( x )\). The curve crosses the axes at \(( p , 0 )\) and \(( 0 , q )\) and the lines \(x = 1\) and \(y = 2\) are asymptotes of the curve.

1. Showing the coordinates of any points of intersection with the axes and the equations of any asymptotes, sketch on separate diagrams the graphs of
   1. \(y = | \mathrm { f } ( x ) |\),
   2. \(y = 2 \mathrm { f } ( x + 1 )\). Given also that $$\mathrm { f } ( x ) \equiv \frac { 2 x - 1 } { x - 1 } , \quad x \in \mathbb { R } , \quad x \neq 1$$
2. find the values of \(p\) and \(q\),
3. find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).

4. The function f is defined by $$\mathrm { f } ( x ) \equiv x ^ { 2 } - 2 a x , \quad x \in \mathbb { R }$$ where \(a\) is a positive constant.

1. Showing the coordinates of any points where the graph meets the axes, sketch the graph of \(y = | \mathrm { f } ( x ) |\). The function \(g\) is defined by $$\mathrm { g } ( x ) \equiv 3 a x , \quad x \in \mathbb { R } .$$
2. Find \(\mathrm { fg } ( \mathrm { a } )\) in terms of \(a\).
3. Solve the equation $$\operatorname { gf } ( x ) = 9 a ^ { 3 }$$

8. \(f ( x ) = x ^ { 2 } - 2 x + 5 , x \in \mathbb { R } , x \geq 1\).

1. Express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
2. State the range of f .
3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
4. Describe fully two transformations that would map the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\) onto the graph of \(y = \sqrt { x } , x \geq 0\).
5. Find an equation for the normal to the curve \(y = \mathrm { f } ^ { - 1 } ( x )\) at the point where \(x = 8\).

5. \(\mathrm { f } ( x ) = 5 + \mathrm { e } ^ { 2 x - 3 } , x \in \mathbb { R }\).

1. State the range of f .
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.
3. Solve the equation \(\mathrm { f } ( x ) = 7\).
4. Find an equation for the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where \(y = 7\).

1. The functions \(f\) and \(g\) are defined by

$$\begin{aligned} & \mathrm { f } : x \rightarrow | 2 x - 5 | , \quad x \in \mathbb { R } , \\ & \mathrm {~g} : x \rightarrow \ln ( x + 3 ) , \quad x \in \mathbb { R } , \quad x > - 3 \end{aligned}$$

1. State the range of f .
2. Evaluate fg(-2).
3. Solve the equation $$\operatorname { fg } ( x ) = 3$$ giving your answers in exact form.
4. Show that the equation $$\mathrm { f } ( x ) = \mathrm { g } ( x )$$ has a root, \(\alpha\), in the interval [3,4].
5. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \left[ 5 + \ln \left( x _ { n } + 3 \right) \right]$$ with \(x _ { 0 } = 3\), to find \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to 4 significant figures.
6. Show that your answer for \(x _ { 4 }\) is the value of \(\alpha\) correct to 4 significant figures.

6. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \rightarrow 1 - a x , \quad x \in \mathbb { R } \\ & \mathrm {~g} : x \rightarrow x ^ { 2 } + 2 a x + 2 , \quad x \in \mathbb { R } \end{aligned}$$ where \(a\) is a constant.
Find, in terms of \(a\),

1. an expression for \(\mathrm { f } ^ { - 1 } ( x )\),
2. the range of g . Given that \(g f ( 3 ) = 7\),
3. find the two possible values of \(a\).

4 \includegraphics[max width=\textwidth, alt={}, center]{d858728a-3371-4755-880c-54f96c5e5156-2_529_737_900_701} The function f is defined by \(\mathrm { f } ( x ) = 2 - \sqrt { x }\) for \(x \geqslant 0\). The graph of \(y = \mathrm { f } ( x )\) is shown above.

1. State the range of f.
2. Find the value of \(\mathrm { ff } ( 4 )\).
3. Given that the equation \(| \mathrm { f } ( x ) | = k\) has two distinct roots, determine the possible values of the constant \(k\).

9 Functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = 2 \sin x & \text { for } - \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi \\ \mathrm {~g} ( x ) = 4 - 2 x ^ { 2 } & \text { for } x \in \mathbb { R } . \end{array}$$

1. State the range of f and the range of g .
2. Show that \(\operatorname { gf } ( 0.5 ) = 2.16\), correct to 3 significant figures, and explain why \(\mathrm { fg } ( 0.5 )\) is not defined.
3. Find the set of values of \(x\) for which \(\mathrm { f } ^ { - 1 } \mathrm {~g} ( x )\) is not defined.