Find inverse function

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

$$\begin{aligned} & \mathrm { f } ( x ) = 2 + 5 \ln x \quad x > 0 \\ & \mathrm {~g} ( x ) = \frac { 6 x - 2 } { 2 x + 1 } \quad x > \frac { 1 } { 3 } \end{aligned}$$

1. Find \(\mathrm { f } ^ { - 1 } ( 22 )\)
2. Use differentiation to prove that g is an increasing function.
3. Find \(\mathrm { g } ^ { - 1 }\)
4. Find the range of fg

2. The functions f and g are defined by $$\begin{array} { l l } f ( x ) = 5 - \frac { 4 } { 3 x + 2 } & x \geqslant 0 \\ g ( x ) = \left| 4 \sin \left( \frac { x } { 3 } + \frac { \pi } { 6 } \right) \right| & x \in \mathbb { R } \end{array}$$

1. Find the range of f
2. 1. Find \(\mathrm { f } ^ { - 1 } ( x )\)
   2. Write down the domain of \(\mathrm { f } ^ { - 1 }\)
3. Find \(\mathrm { fg } ( - \pi )\)

1. The functions f and g are defined by

$$\mathrm { f } : x \mapsto \mathrm { e } ^ { x } + 2 \quad x \in \mathbb { R }$$ $$\mathrm { g } : x \mapsto \ln x \quad x > 0$$

1. State the range of f .
2. Find \(\mathrm { fg } ( x )\), giving \(y\) our answer in its simplest form.
3. Find the exact value of \(x\) for which \(\mathrm { f } ( 2 x + 3 ) = 6\)
4. Find \(\mathrm { f } ^ { - 1 }\) stating its domain.
5. On the same axes sketch the curves with equation \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), giving the coordinates of all the points where the curves cross the axes.

1. It is given that

$$\begin{gathered} \mathrm { f } ( x ) = \mathrm { e } ^ { - 2 x } \quad x \in \mathbb { R } \\ \mathrm {~g} ( x ) = \frac { x } { x - 3 } \quad x > 3 \end{gathered}$$

1. Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of any points where the graph crosses the axes.
2. Find the range of g
3. Find \(\mathrm { g } ^ { - 1 } ( x )\), stating the domain of \(\mathrm { g } ^ { - 1 }\)
4. Using algebra, find the exact value of \(x\) for which \(\operatorname { fg } ( x ) = 3\)

3. The function g is defined by $$g ( x ) = \frac { 6 x } { 2 x + 3 } \quad x > 0$$

1. Find the range of g .
2. Find \(\mathrm { g } ^ { - 1 } ( x )\) and state its domain.
3. Find the function \(\operatorname { gg } ( x )\), writing your answer as a single fraction in its simplest form.

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

$$\begin{array} { l l } \mathrm { f } : x \rightarrow \mathrm { e } ^ { 2 x } - 5 , & x \in \mathbb { R } \\ \mathrm {~g} : x \rightarrow \ln ( 3 x - 1 ) , & x \in \mathbb { R } , x > \frac { 1 } { 3 } \end{array}$$

1. Find \(\mathrm { f } ^ { - 1 }\) and state its domain.
2. Find \(\mathrm { fg } ( 3 )\), giving your answer in its simplest form.
   (ii) (a) Sketch the graph with equation $$y = | 4 x - a |$$ where \(a\) is a positive constant. State the coordinates of each point where the graph cuts or meets the coordinate axes. Given that $$| 4 x - a | = 9 a$$ where \(a\) is a positive constant,
3. find the possible values of $$| x - 6 a | + 3 | x |$$ giving your answers, in terms of \(a\), in their simplest form.

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

$$\begin{array} { l l } \mathrm { f } : x \mapsto \mathrm { e } ^ { - x } + 2 , & x \in \mathbb { R } \\ \mathrm {~g} : x \mapsto 2 \ln x , & x > 0 \end{array}$$

1. Find \(\mathrm { fg } ( x )\), giving your answer in its simplest form.
2. Find the exact value of \(x\) for which \(\mathrm { f } ( 2 x + 3 ) = 6\)
3. Find \(\mathrm { f } ^ { - 1 }\), stating its domain.
4. On the same axes, sketch the curves with equation \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), giving the coordinates of all the points where the curves cross the axes.

1. The function \(f\) is defined by

$$\mathrm { f } : x \mapsto \ln ( 4 - 2 x ) , \quad x < 2 \quad \text { and } \quad x \in \mathbb { R } .$$

1. Show that the inverse function of f is defined by $$\mathrm { f } ^ { - 1 } : x \mapsto 2 - \frac { 1 } { 2 } \mathrm { e } ^ { x }$$ and write down the domain of \(\mathrm { f } ^ { - 1 }\).
2. Write down the range of \(\mathrm { f } ^ { - 1 }\).
3. In the space provided on page 16, sketch the graph of \(y = f ^ { - 1 } ( x )\). State the coordinates of the points of intersection with the \(x\) and \(y\) axes. The graph of \(y = x + 2\) crosses the graph of \(y = f ^ { - 1 } ( x )\) at \(x = k\). The iterative formula $$x _ { n + 1 } = - \frac { 1 } { 2 } e ^ { x _ { n } } , x _ { 0 } = - 0.3$$ is used to find an approximate value for \(k\).
4. Calculate the values of \(x _ { 1 }\) and \(x _ { 2 }\), giving your answers to 4 decimal places.
5. Find the value of \(k\) to 3 decimal places.

1. The function \(f\) is defined by

$$\mathrm { f } : x \mapsto \frac { 3 - 2 x } { x - 5 } , \quad x \in \mathbb { R } , x \neq 5$$

1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
   (3) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3ff6824f-9fbf-4b5b-8bab-91332c549b36-10_901_1091_593_429} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} The function g has domain \(- 1 \leqslant x \leqslant 8\), and is linear from \(( - 1 , - 9 )\) to \(( 2,0 )\) and from \(( 2,0 )\) to \(( 8,4 )\). Figure 2 shows a sketch of the graph of \(y = \mathrm { g } ( x )\).
2. Write down the range of g.
3. Find \(\operatorname { gg } ( 2 )\).
4. Find \(\mathrm { fg } ( 8 )\).
5. On separate diagrams, sketch the graph with equation
   1. \(y = | \mathrm { g } ( x ) |\),
   2. \(y = \mathrm { g } ^ { - 1 } ( x )\). Show on each sketch the coordinates of each point at which the graph meets or cuts the axes.
6. State the domain of the inverse function \(\mathrm { g } ^ { - 1 }\).

5. The functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto \ln ( 2 x - 1 ) , & x \in \mathbb { R } , x > \frac { 1 } { 2 } \\ \mathrm {~g} : x \mapsto \frac { 2 } { x - 3 } , & x \in \mathbb { R } , x \neq 3 \end{array}$$

1. Find the exact value of fg(4).
2. Find the inverse function \(\mathrm { f } ^ { - 1 } ( x )\), stating its domain.
3. Sketch the graph of \(y = | \mathrm { g } ( x ) |\). Indicate clearly the equation of the vertical asymptote and the coordinates of the point at which the graph crosses the \(y\)-axis.
4. Find the exact values of \(x\) for which \(\left| \frac { 2 } { x - 3 } \right| = 3\).

4. The function \(f\) is defined by $$f : x \mapsto \frac { 2 ( x - 1 ) } { x ^ { 2 } - 2 x - 3 } - \frac { 1 } { x - 3 } , \quad x > 3$$

1. Show that \(\mathrm { f } ( x ) = \frac { 1 } { x + 1 } , \quad x > 3\).
2. Find the range of f.
3. Find \(\mathrm { f } ^ { - 1 } ( x )\). State the domain of this inverse function. The function \(g\) is defined by $$\mathrm { g } : x \mapsto 2 x ^ { 2 } - 3 , \quad x \in \mathbb { R }$$
4. Solve \(\mathrm { fg } ( x ) = \frac { 1 } { 8 }\).

4. The function \(f\) is defined by $$\mathrm { f } : x \mapsto 4 - \ln ( x + 2 ) , \quad x \in \mathbb { R } , x \geqslant - 1$$

1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
2. Find the domain of \(\mathrm { f } ^ { - 1 }\). The function \(g\) is defined by $$\mathrm { g } : x \mapsto \mathrm { e } ^ { x ^ { 2 } } - 2 , \quad x \in \mathbb { R }$$
3. Find \(\mathrm { fg } ( x )\), giving your answer in its simplest form.
4. Find the range of fg.

6. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto \mathrm { e } ^ { x } + 2 , \quad x \in \mathbb { R } \\ & \mathrm {~g} : x \mapsto \ln x , \quad x > 0 \end{aligned}$$

1. State the range of f.
2. Find \(\mathrm { fg } ( x )\), giving your answer in its simplest form.
3. Find the exact value of \(x\) for which \(\mathrm { f } ( 2 x + 3 ) = 6\)
4. Find \(\mathrm { f } ^ { - 1 }\), the inverse function of f , stating its domain.
5. On the same axes sketch the curves with equation \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), giving the coordinates of all the points where the curves cross the axes.

6. The function f is defined by $$\mathrm { f } : x \rightarrow \mathrm { e } ^ { 2 x } + k ^ { 2 } , \quad x \in \mathbb { R } , \quad k \text { is a positive constant. }$$

1. State the range of f .
2. Find \(\mathrm { f } ^ { - 1 }\) and state its domain. The function g is defined by $$g : x \rightarrow \ln ( 2 x ) , \quad x > 0$$
3. Solve the equation $$\mathrm { g } ( x ) + \mathrm { g } \left( x ^ { 2 } \right) + \mathrm { g } \left( x ^ { 3 } \right) = 6$$ giving your answer in its simplest form.
4. Find \(\mathrm { fg } ( x )\), giving your answer in its simplest form.
5. Find, in terms of the constant \(k\), the solution of the equation $$\mathrm { fg } ( x ) = 2 k ^ { 2 }$$

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 $$\mathrm { f } ^ { - 1 } \mathrm {~g} ( x ) = 4$$

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.

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 \(y = 7\).

6 \includegraphics[max width=\textwidth, alt={}, center]{ebfdf170-99c6-4785-b9d7-201c3425b4c9-3_563_583_267_781} The diagram shows the graph of \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = 2 - x ^ { 2 } , \quad x \leqslant 0 .$$

1. Evaluate \(\mathrm { ff } ( - 3 )\).
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
3. Sketch the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\). Indicate the coordinates of the points where the graph meets the axes.

3 The function \(f\) is defined for all non-negative values of \(x\) by $$f ( x ) = 3 + \sqrt { x }$$

1. Evaluate ff(169).
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( \mathrm { x } )\) in terms of x .
3. On a single diagram sketch the graphs of \(y = f ( x )\) and \(y = f ^ { - 1 } ( x )\), indicating how the two graphs are related.

6 The function \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 1 + 2 \sin x\) for \(- \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi\).

1. Show that \(\mathrm { f } ^ { - 1 } ( x ) = \arcsin \left( \frac { x - 1 } { 2 } \right)\) and state the domain of this function. Fig. 6 shows a sketch of the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{6a4c3f3b-a298-4b13-b97e-b52f8d9d527b-4_506_561_705_751} \captionsetup{labelformat=empty} \caption{Fig. 6}
   \end{figure}
2. Write down the coordinates of the points \(\mathrm { A } , \mathrm { B }\) and C .

6 The function \(\mathrm { f } ( x )\) is defined by $$f ( x ) = 1 + 2 \sin 3 x , \quad - \frac { \pi } { 6 } \leqslant x \leqslant \frac { \pi } { 6 }$$ You are given that this function has an inverse, \(\mathrm { f } ^ { - 1 } ( x )\).
Find \(\mathrm { f } ^ { - 1 } ( x )\) and its domain.

2 Fig. 6 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 1 } { 2 } \arctan x\). \begin{figure}[h] \end{figure}

1. Find the range of the function \(\mathrm { f } ( x )\), giving your answer in terms of \(\pi\).
2. Find the inverse function \(\mathrm { f } ^ { - 1 } ( x )\). Find the gradient of the curve \(y = \mathrm { f } ^ { - 1 } ( x )\) at the origin.
3. Hence write down the gradient of \(y = \frac { 1 } { 2 } \arctan x\) at the origin.

1 Fig. 4 shows the curve \(y = \mathrm { f } ( x )\), where $$f ( x ) = a + \cos b x , 0 \leqslant x \leqslant 2 \pi ,$$ and \(a\) and \(b\) are positive constants. The curve has stationary points at \(( 0,3 )\) and \(( 2 \pi , 1 )\). \begin{figure}[h] \end{figure}

1. Find \(a\) and \(b\).
2. Find \(\mathrm { f } ^ { - 1 } ( x )\), and state its domain and range.