Verify composite identity

9 Functions f, g and h are defined as follows: $$\begin{aligned} & \mathrm { f } : x \mapsto x - 4 x ^ { \frac { 1 } { 2 } } + 1 \quad \text { for } x \geqslant 0 \\ & \mathrm {~g} : x \mapsto m x ^ { 2 } + n \quad \text { for } x \geqslant - 2 , \text { where } m \text { and } n \text { are constants, } \\ & \mathrm { h } : x \mapsto x ^ { \frac { 1 } { 2 } } - 2 \quad \text { for } x \geqslant 0 . \end{aligned}$$

1. Solve the equation \(\mathrm { f } ( x ) = 0\), giving your solutions in the form \(x = a + b \sqrt { c }\), where \(a , b\) and \(c\) are integers.
2. Given that \(\mathrm { f } ( x ) \equiv \mathrm { gh } ( x )\), find the values of \(m\) and \(n\). \includegraphics[max width=\textwidth, alt={}, center]{05e75fa2-81ae-44b1-b073-4100f5d911e0-16_652_1045_255_550} The diagram shows a circle with centre \(A\) of radius 5 cm and a circle with centre \(B\) of radius 8 cm . The circles touch at the point \(C\) so that \(A C B\) is a straight line. The tangent at the point \(D\) on the smaller circle intersects the larger circle at \(E\) and passes through \(B\).

7. The function f is defined by $$\mathrm { f } : x \mapsto \frac { 3 x - 5 } { x + 1 } , \quad x \in \mathbb { R } , x \neq - 1$$

1. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\)
2. Show that $$\operatorname { ff } ( x ) = \frac { x + a } { x - 1 } , \quad x \in \mathbb { R } , x \neq - 1 , x \neq 1$$ where \(a\) is an integer to be determined. The function \(g\) is defined by $$\mathrm { g } : x \mapsto x ^ { 2 } - 3 x , \quad x \in \mathbb { R } , 0 \leqslant x \leqslant 5$$
3. Find the value of fg(2)
4. Find the range of g

8. The functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } : x \rightarrow 2 x + \ln 2 , & x \in \mathbb { R } , \\ \mathrm {~g} : x \rightarrow \mathrm { e } ^ { 2 x } , & x \in \mathbb { R } . \end{array}$$

1. Prove that the composite function gf is $$\operatorname { gf } : x \rightarrow 4 \mathrm { e } ^ { 4 x } , \quad x \in \mathbb { R }$$
2. In the space provided on page 19, sketch the curve with equation \(y = \operatorname { gf } ( x )\), and show the coordinates of the point where the curve cuts the \(y\)-axis.
3. Write down the range of gf.
4. Find the value of \(x\) for which \(\frac { \mathrm { d } } { \mathrm { d } x } [ \operatorname { gf } ( x ) ] = 3\), giving your answer to 3 significant figures.

5. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 3 x + \ln x , \quad x > 0 , \quad x \in \mathbb { R } \\ & \mathrm {~g} : x \mapsto \mathrm { e } ^ { x ^ { 2 } } , \quad x \in \mathbb { R } \end{aligned}$$

1. Write down the range of g.
2. Show that the composite function fg is defined by $$\mathrm { fg } : x \mapsto x ^ { 2 } + 3 \mathrm { e } ^ { x ^ { 2 } } , \quad x \in \mathbb { R } .$$
3. Write down the range of fg.
4. Solve the equation \(\frac { \mathrm { d } } { \mathrm { d } x } [ \mathrm { fg } ( x ) ] = x \left( x \mathrm { e } ^ { x ^ { 2 } } + 2 \right)\).

2 Fig. 9 shows the line \(y = x\) and the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } - 1 \right)\). The line and the curve intersect at the origin and at the point \(\mathrm { P } ( a , a )\). \begin{figure}[h] \end{figure}

1. Show that \(\mathrm { e } ^ { a } = 1 + 2 a\).
2. Show that the area of the region enclosed by the curve, the \(x\)-axis and the line \(x = a\) is \(\frac { 1 } { 2 } a\). Hence find, in terms of \(a\), the area enclosed by the curve and the line \(y = x\).
3. Show that the inverse function of \(\mathrm { f } ( x )\) is \(\mathrm { g } ( x )\), where \(\mathrm { g } ( x ) = \ln ( 1 + 2 x )\). Add a sketch of \(y = \mathrm { g } ( x )\) to the copy of Fig. 9.
4. Find the derivatives of \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\). Hence verify that \(\mathrm { g } ^ { \prime } ( a ) = \frac { 1 } { \mathrm { f } ^ { \prime } ( a ) }\). Give a geometrical interpretation of this result.

1 Given that \(\mathrm { f } ( x ) = \frac { x + 1 } { x - 1 }\), show that \(\mathrm { ff } ( x ) = x\).
Hence write down the inverse function \(\mathrm { f } ^ { - 1 } ( x )\). What can you deduce about the symmetry of the curve \(y = \mathrm { f } ( x ) ?\)

2 Fig. 8 shows the line \(y = x\) and parts of the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\), where $$\mathrm { f } ( x ) = \mathrm { e } ^ { x - 1 } , \quad \mathrm {~g} ( x ) = 1 + \ln x$$ The curves intersect the axes at the points A and B, as shown. The curves and the line \(y = x\) meet at the point C . \begin{figure}[h] \end{figure}

1. Find the exact coordinates of A and B . Verify that the coordinates of C are \(( 1,1 )\).
2. Prove algebraically that \(\mathrm { g } ( x )\) is the inverse of \(\mathrm { f } ( x )\).
3. Evaluate \(\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x\), giving your answer in terms of e .
4. Use integration by parts to find \(\int \ln x \mathrm {~d} x\). Hence show that \(\int _ { \mathrm { e } ^ { - 1 } } ^ { 1 } \mathrm {~g} ( x ) \mathrm { d } x = \frac { 1 } { \mathrm { e } }\).
5. Find the area of the region enclosed by the lines OA and OB , and the arcs AC and BC .

3 Given that \(\mathrm { f } ( x ) = \frac { 1 } { 2 } \ln ( x - 1 )\) and \(\mathrm { g } ( x ) = 1 + \mathrm { e } ^ { 2 x }\), show that \(\mathrm { g } ( x )\) is the inverse of \(\mathrm { f } ( x )\).

6 Write down the conditions for \(\mathrm { f } ( x )\) to be an odd function and for \(\mathrm { g } ( x )\) to be an even function.
Hence prove that, if \(\mathrm { f } ( x )\) is odd and \(\mathrm { g } ( x )\) is even, then the composite function \(\mathrm { gf } ( x )\) is even.

6 Given that \(\mathrm { f } ( x ) = \frac { x + 1 } { x - 1 }\), show that \(\mathrm { ff } ( x ) = x\).
Hence write down the inverse function \(\mathrm { f } ^ { - 1 } ( x )\). What can you deduce about the symmetry of the curve \(y = \mathrm { f } ( x )\) ?

8 Fig. 8 shows the line \(y = x\) and parts of the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\), where $$\mathrm { f } ( x ) = \mathrm { e } ^ { x - 1 } , \quad \mathrm {~g} ( x ) = 1 + \ln x$$ The curves intersect the axes at the points A and B , as shown. The curves and the line \(y = x\) meet at the point C . \begin{figure}[h] \end{figure}

1. Find the exact coordinates of A and B . Verify that the coordinates of C are \(( 1,1 )\).
2. Prove algebraically that \(\mathrm { g } ( x )\) is the inverse of \(\mathrm { f } ( x )\).
3. Evaluate \(\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x\), giving your answer in terms of e.
4. Use integration by parts to find \(\int \ln x \mathrm {~d} x\). Hence show that \(\int _ { \mathrm { e } ^ { - 1 } } ^ { 1 } \mathrm {~g} ( x ) \mathrm { d } x = \frac { 1 } { \mathrm { e } }\).
5. Find the area of the region enclosed by the lines OA and OB , and the arcs AC and BC .

7

1. The function \(\mathrm { f } ( x )\) is defined by $$f ( x ) = \frac { 1 - x } { 1 + x } , x \neq - 1$$ Show that \(\mathrm { f } ( \mathrm { f } ( x ) ) = x\).
   Hence write down \(\mathrm { f } ^ { - 1 } ( x )\).
2. The function \(\mathrm { g } ( x )\) is defined for all real \(x\) by $$\mathrm { g } ( x ) = \frac { 1 - x ^ { 2 } } { 1 + x ^ { 2 } }$$ Prove that \(\mathrm { g } ( x )\) is even. Interpret this result in terms of the graph of \(y = \mathrm { g } ( x )\).

1. The function f is defined by

$$f ( x ) = \frac { 3 x - 7 } { x - 2 } \quad x \in \mathbb { R } , x \neq 2$$

1. Find \(f ^ { - 1 } ( 7 )\)
2. Show that \(\operatorname { ff } ( x ) = \frac { a x + b } { x - 3 }\) where \(a\) and \(b\) are integers to be found.

   VI4V SIHI NI ILIUM ION OC

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

1. Prove that the composite function gf is $$\operatorname { gf } : x \mapsto 4 \mathrm { e } ^ { 4 x } , \quad x \in \mathbb { R }$$
2. Sketch the curve with equation \(y = \operatorname { gf } ( x )\), and show the coordinates of the point where the curve cuts the \(y\)-axis.
3. Write down the range of gf .
4. Find the value of \(x\) for which \(\frac { \mathrm { d } } { \mathrm { d } x } [ \operatorname { gf } ( x ) ] = 3\), giving your answer to 3 significant figures.

The function \(f\) is defined by \(f : x \mapsto \frac{x + 3}{2x - 1}\), \(x \in \mathbb{R}\), \(x \neq \frac{1}{2}\).

1. Show that \(f f(x) = x\). [3]
2. Hence, or otherwise, obtain an expression for \(f^{-1}(x)\). [2]

The function \(f\) is defined by \(f: x \mapsto \frac{3x-1}{x-3}, x \in \mathbb{R}, x \neq 3\).

1. Prove that \(f^{-1}(x) = f(x)\) for all \(x \in \mathbb{R}, x \neq 3\). [3]
2. Hence find, in terms of \(k\), \(ff(k)\), where \(x \neq 3\). [2]

\includegraphics{figure_3} Figure 3 shows a sketch of the one-one function \(g\), defined over the domain \(-2 \leq x \leq 2\).

1. Find the value of \(fg(-2)\). [3]
2. Sketch the graph of the inverse function \(g^{-1}\) and state its domain. [3]

The function \(h\) is defined by \(h: x \mapsto 2g(x - 1)\).

1. Sketch the graph of the function \(h\) and state its range. [3]

The function f is defined by \(\text{f}: x \to \frac{3x - 1}{x - 3}\), \(x \in \mathbb{R}\), \(x \neq 3\).

1. Prove that \(\text{f}^{-1}(x) = \text{f}(x)\) for all \(x \in \mathbb{R}\), \(x \neq 3\). [3]
2. Hence find, in terms of \(k\), \(\text{f}f(k)\), where \(x \neq 3\). [2]

\includegraphics{figure_3} Figure 3 shows a sketch of the one-one function g, defined over the domain \(-2 \leq x \leq 2\).

1. Find the value of \(\text{f}g(-2)\). [3]
2. Sketch the graph of the inverse function \(\text{g}^{-1}\) and state its domain. [3]

The function h is defined by \(\text{h}: x \mapsto 2g(x - 1)\).

1. Sketch the graph of the function h and state its range. [3]

Fig. 9 shows the line \(y = x\) and the curve \(y = f(x)\), where \(f(x) = \frac{1}{2}(e^x - 1)\). The line and the curve intersect at the origin and at the point P\((a, a)\). \includegraphics{figure_9}

1. Show that \(e^a = 1 + 2a\). [1]
2. Show that the area of the region enclosed by the curve, the \(x\)-axis and the line \(x = a\) is \(\frac{1}{2}a\). Hence find, in terms of \(a\), the area enclosed by the curve and the line \(y = x\). [6]
3. Show that the inverse function of f\((x)\) is g\((x)\), where g\((x) = \ln(1 + 2x)\). Add a sketch of \(y = g(x)\) to the copy of Fig. 9. [5]
4. Find the derivatives of f\((x)\) and g\((x)\). Hence verify that \(g'(a) = \frac{1}{f'(a)}\). Give a geometrical interpretation of this result. [7]

\(f(x) \equiv \frac{2x-3}{x-2}\), \(x \in \mathbb{R}\), \(x > 2\).

1. Find the range of \(f\). [2]
2. Show that \(f(f(x) = x\) for all \(x > 2\). [3]
3. Hence, write down an expression for \(f^{-1}(x)\). [1]

You are given that f(x) and g(x) are odd functions, defined for \(x \in \mathbb{R}\).

1. Given that s(x) = f(x) + g(x), prove that s(x) is an odd function. [2]
2. Given that p(x) = f(x)g(x), determine whether p(x) is odd, even or neither. [2]

Write down the conditions for f(x) to be an odd function and for g(x) to be an even function. Hence prove that, if f(x) is odd and g(x) is even, then the composite function gf(x) is even. [4]

Given that \(f(x) = \frac{1}{2}\ln(x - 1)\) and \(g(x) = 1 + e^{2x}\), show that g(x) is the inverse of f(x). [3]

The function \(f\) is defined by $$f(x) = \frac{5x}{7x - 5}$$

1. The domain of \(f\) is the set \(\{x \in \mathbb{R} : x \neq a\}\) State the value of \(a\) [1 mark]
2. Prove that \(f\) is a self-inverse function [3 marks]
3. Find the range of \(f\) [1 mark]