Composite & Inverse Functions

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.

The function f is defined by \(f(x) = e^x + 1\) for \(x \in \mathbb{R}\) Find an expression for \(f^{-1}(x)\) Tick \((\checkmark)\) one box. [1 mark] \(f^{-1}(x) = \ln(x - 1)\) \(\square\) \(f^{-1}(x) = \ln(x) - 1\) \(\square\) \(f^{-1}(x) = \frac{1}{e^x + 1}\) \(\square\) \(f^{-1}(x) = \frac{x - 1}{e}\) \(\square\)

$$f(x) = \frac{ax + b}{x + 2}; \quad x \in \mathbb{R}, x \neq -2,$$ where \(a\) and \(b\) are constants and \(b > 0\).

1. Find \(f^{-1}(x)\). [2]
2. Hence, or otherwise, find the value of \(a\) so that \(f(x) = x\). [2]

The curve \(C\) has equation \(y = f(x)\) and \(f(x)\) satisfies \(f(x) = x\).

1. On separate axes sketch
   1. \(y = f(x)\), [3]
   2. \(y = f(x - 2) + 2\). [3]

On each sketch you should indicate the equations of any asymptotes and the coordinates, in terms of \(b\), of any intersections with the axes. The normal to \(C\) at the point \(P\) has equation \(y = 4x - 39\). The normal to \(C\) at the point \(Q\) has equation \(y = 4x + k\), where \(k\) is a constant.

1. By considering the images of the normals to \(C\) on the curve with equation \(y = f(x - 2) + 2\), or otherwise, find the value of \(k\). [5]

Given that \(f(x) = 2\ln x\) and \(g(x) = e^x\), find the composite function gf(x), expressing your answer as simply as possible. [3]

7 The functions \(f , g\) and \(h\) are defined as follows. $$\mathrm { f } ( x ) = 2 x \quad \mathrm {~g} ( x ) = x ^ { 2 } \quad \mathrm {~h} ( x ) = x + 2$$ Find each of the following as functions of \(x\).

1. \(\mathrm { f } ^ { 2 } ( x )\),
2. \(\operatorname { fgh } ( x )\),
3. \(\mathrm { h } ^ { - 1 } ( x )\).

% Figure 4 shows curves with asymptotic behavior at x = 3 \includegraphics{figure_4} Figure 4

1. Figure 4 shows a sketch of the curve with equation \(y = f(x)\), where $$f(x) = \frac{x^2 - 5}{3-x}, \quad x \in \mathbb{R}, x \neq 3$$ The curve has a minimum at the point \(A\), with \(x\)-coordinate \(\alpha\), and a maximum at the point \(B\), with \(x\)-coordinate \(\beta\). Find the value of \(\alpha\), the value of \(\beta\) and the \(y\)-coordinates of the points \(A\) and \(B\). [5]
2. The functions \(g\) and \(h\) are defined as follows $$g: x \to x + p \quad x \in \mathbb{R}$$ $$h: x \to |x| \quad x \in \mathbb{R}$$ where \(p\) is a constant. % Figure 5 shows curve with minimum points at C and D symmetric about y-axis \includegraphics{figure_5} Figure 5 Figure 5 shows a sketch of the curve with equation \(y = h(fg(x) + q)\), \(x \in \mathbb{R}\), \(x \neq 0\), where \(q\) is a constant. The curve is symmetric about the \(y\)-axis and has minimum points at \(C\) and \(D\).
   1. Find the value of \(p\) and the value of \(q\).
   2. Write down the coordinates of \(D\).
   [5]
3. The function \(\mathrm{m}\) is given by $$\mathrm{m}(x) = \frac{x^2 - 5}{3-x} \quad x \in \mathbb{R}, x < \alpha$$ where \(\alpha\) is the \(x\)-coordinate of \(A\) as found in part (a).
   1. Find \(\mathrm{m}^{-1}\)
   2. Write down the domain of \(\mathrm{m}^{-1}\)
   3. Find the value of \(t\) such that \(\mathrm{m}(t) = \mathrm{m}^{-1}(t)\)
   [10]

[Total 20 marks]

The functions \(f(x)\) and \(g(x)\) are defined by \(f(x) = \ln x\) and \(g(x) = 2 + e^x\), for \(x > 0\). Find the exact value of \(x\), given that \(fg(x) = 2x\). [5]

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 \(\mathrm { ff } ( x ) = \mathrm { gf } ( 2 )\).

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).

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]

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.

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.

Determine which one of these graphs does not represent \(y\) as a function of \(x\). Tick (\(\checkmark\)) one box. [1 mark] \includegraphics{figure_3}

Determine which one of these graphs does not represent \(y\) as a function of \(x\). Tick (\(\checkmark\)) one box. [1 mark] \includegraphics{figure_3}

10 The function h is defined by $$\mathrm { h } ( x ) = \frac { \sqrt { x } } { x - 3 }$$ where \(h\) has its maximum possible domain.
10

1. Find the domain of h .
   Give your answer using set notation. 10
2. Alice correctly calculates $$h ( 1 ) = - 0.5 \text { and } h ( 4 ) = 2$$ She then argues that since there is a change of sign there must be a value of \(x\) in the interval \(1 < x < 4\) that gives \(\mathrm { h } ( x ) = 0\) Explain the error in Alice's argument.
   [0pt] [2 marks]
   10
3. By considering any turning points of h , determine whether h has an inverse function. Fully justify your answer.
   [0pt] [6 marks]

1 The function f is defined for all real values of \(x\) by $$f ( x ) = 10 - ( x + 3 ) ^ { 2 } .$$

1. State the range of f .
2. Find the value of \(\mathrm { ff } ( - 1 )\).

1 Functions f and g are defined for all real values of \(x\) by $$\mathrm { f } ( x ) = x ^ { 3 } + 4 \quad \text { and } \quad \mathrm { g } ( x ) = 2 x - 5$$ Evaluate

1. \(f g ( 1 )\),
2. \(\mathrm { f } ^ { - 1 } ( 12 )\).

7. The function \(f\) has domain \(- 2 \leqslant x \leqslant 6\) and is linear from \(( - 2,10 )\) to \(( 2,0 )\) and from \(( 2,0 )\) to (6, 4). A sketch of the graph of \(y = \mathrm { f } ( x )\) is shown in Figure 1. \begin{figure}[h] \end{figure}

1. Write down the range of f .
2. Find \(\mathrm { ff } ( 0 )\). The function \(g\) is defined by $$\mathrm { g } : x \rightarrow \frac { 4 + 3 x } { 5 - x } , \quad x \in \mathbb { R } , \quad x \neq 5$$
3. Find \(\mathrm { g } ^ { - 1 } ( x )\)
4. Solve the equation \(\operatorname { gf } ( x ) = 16\)

2 The function f is defined by \(\mathrm { f } ( x ) = - 2 x ^ { 2 } - 8 x - 13\) for \(x < - 3\).

1. Express \(\mathrm { f } ( x )\) in the form \(- 2 ( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are integers.
2. Find the range of f.
3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).

2 The function f is defined by \(\mathrm { f } ( x ) = - 2 x ^ { 2 } - 8 x - 13\) for \(x < - 3\).

1. Express \(\mathrm { f } ( x )\) in the form \(- 2 ( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are integers.
2. Find the range of f.
3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).

9

1. Express \(2 x ^ { 2 } + 12 x + 11\) in the form \(2 ( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
   The function f is defined by \(\mathrm { f } ( x ) = 2 x ^ { 2 } + 12 x + 11\) for \(x \leqslant - 4\).
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
   The function g is defined by \(\mathrm { g } ( x ) = 2 x - 3\) for \(x \leqslant k\).
3. For the case where \(k = - 1\), solve the equation \(\operatorname { fg } ( x ) = 193\).
4. State the largest value of \(k\) possible for the composition fg to be defined.

The function f(x) is defined by \(\text{f}(x) = x^3 - 4\) for \(-1 \leq x \leq 2\). For \(\text{f}^{-1}(x)\), determine

• the domain
• the range.

[5]

It is given that \(y = \frac{1}{x}\) and \(x < -1\) Determine which statement below fully describes the possible values of \(y\). Tick (\(\checkmark\)) one box. [1 mark] \(-\infty < y < -1\) \(y > -1\) \(-1 < y < 0\) \(y < 0\)

3. The function f is defined by $$f : x \mapsto 2 x ^ { 2 } + 3 k x + k ^ { 2 } \quad x \in \mathbb { R } , - 4 k \leqslant x \leqslant 0$$ where \(k\) is a positive constant.

1. Find, in terms of \(k\), the range of f . The function g is defined by $$\mathrm { g } : x \mapsto 2 k - 3 x \quad x \in \mathbb { R }$$ Given that \(\operatorname { gf } ( - 2 ) = - 12\)
2. find the possible values of \(k\).

The function f is defined by $$f(x) = 2x + 1$$ Solve the equation $$f(x) = f^{-1}(x)$$ Circle your answer. [1 mark] \(x = -1\) \quad\quad \(x = 0\) \quad\quad \(x = 1\) \quad\quad \(x = 2\)

The function f is defined by $$f(x) = 2x + 1$$ Solve the equation $$f(x) = f^{-1}(x)$$ Circle your answer. [1 mark] \(x = -1\) \quad\quad \(x = 0\) \quad\quad \(x = 1\) \quad\quad \(x = 2\)

8. The function \(f\) is defined by $$\mathrm { f } : x \rightarrow 3 - 2 \mathrm { e } ^ { - x } , \quad x \in \mathbb { R }$$

1. Find the inverse function, \(\mathrm { f } ^ { - 1 } ( x )\) and give its domain.
2. Solve the equation \(\mathrm { f } ^ { - 1 } ( x ) = \ln x\). The equation \(\mathrm { f } ( t ) = k \mathrm { e } ^ { t }\), where \(k\) is a positive constant, has exactly one real solution.
3. Find the value of \(k\).

10 The one-one function f is defined by \(\mathrm { f } ( x ) = ( x - 2 ) ^ { 2 } + 2\) for \(x \geqslant c\), where \(c\) is a constant.

1. State the smallest possible value of \(c\).
   In parts (ii) and (iii) the value of \(c\) is 4 .
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
3. Solve the equation \(\mathrm { ff } ( x ) = 51\), giving your answer in the form \(a + \sqrt { } b\).

10 The one-one function f is defined by \(\mathrm { f } ( x ) = ( x - 2 ) ^ { 2 } + 2\) for \(x \geqslant c\), where \(c\) is a constant.

1. State the smallest possible value of \(c\).
   In parts (ii) and (iii) the value of \(c\) is 4 .
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
3. Solve the equation \(\mathrm { ff } ( x ) = 51\), giving your answer in the form \(a + \sqrt { } b\).

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 \(\mathrm { g } ^ { - 1 } ( x )\).

The functions f and g are defined for all real values of \(x\) by $$f(x) = 3x - 2 \quad \text{and} \quad g(x) = 3x + 7.$$ Find the exact coordinates of the point at which

1. the graph of \(y = fg(x)\) meets the \(x\)-axis, [3]
2. the graph of \(y = g(x)\) meets the graph of \(y = g^{-1}(x)\), [3]
3. the graph of \(y = |f(x)|\) meets the graph of \(y = |g(x)|\). [4]

The functions f and g are defined for all real values of \(x\) by $$f(x) = 3x - 2 \quad \text{and} \quad g(x) = 3x + 7.$$ Find the exact coordinates of the point at which

1. the graph of \(y = fg(x)\) meets the \(x\)-axis, [3]
2. the graph of \(y = g(x)\) meets the graph of \(y = g^{-1}(x)\), [3]
3. the graph of \(y = |f(x)|\) meets the graph of \(y = |g(x)|\). [4]

11 The function f is defined by \(\mathrm { f } ( x ) = 10 + 6 x - x ^ { 2 }\) for \(x \in \mathbb { R }\).

1. By completing the square, find the range of f . \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-16_2715_37_143_2010} The function g is defined by \(\mathrm { g } ( x ) = 4 x + k\) for \(x \in \mathbb { R }\) where \(k\) is a constant.
2. It is given that the graph of \(y = \mathrm { g } ^ { - 1 } \mathrm { f } ( x )\) meets the graph of \(y = \mathrm { g } ( x )\) at a single point \(P\). Determine the coordinates of \(P\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown. \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-18_2715_35_143_2012}

Write down the inverse function of \(y = e^x\). On the same set of axes, sketch the graphs of \(y = e^x\) and its inverse function, clearly labelling the coordinates of the points where the graphs cross the \(x\) and \(y\) axes. [3]

Write down the inverse function of \(y = e^x\). On the same set of axes, sketch the graphs of \(y = e^x\) and its inverse function, clearly labelling the coordinates of the points where the graphs cross the \(x\) and \(y\) axes. [3]

% Figure shows a curve with maximum at point A, passing through origin O, with horizontal asymptote \includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = f(x)\) where $$f(x) = \frac{x^2 + 16}{3x} \quad x \neq 0$$ The curve has a maximum at the point \(A\) with coordinates \((a, b)\).

1. Find the value of \(a\) and the value of \(b\). [4] The function g is defined as $$g : x \mapsto \frac{x^2 + 16}{3x} \quad a \leq x < 0$$ where \(a\) is the value found in part (a).
2. Write down the range of g. [1]
3. On the same axes sketch \(y = g(x)\) and \(y = g^{-1}(x)\). [3]
4. Find an expression for \(g^{-1}(x)\) and state the domain of \(g^{-1}\) [5]
5. Solve the equation \(g(x) = g^{-1}(x)\). [3]

1 A curve for which \(y\) is inversely proportional to \(x\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{c30a926b-d832-46f5-aa65-0066ef482c3d-4_824_1125_561_242} Find the equation of the curve.

2 The function \(\mathrm { f } ( x ) = \sqrt { x }\) is defined on the domain \(x \geqslant 0\).
The function \(\mathrm { g } ( x ) = 25 - x ^ { 2 }\) is defined on the domain \(\mathbb { R }\).

1. Write down an expression for \(\mathrm { fg } ( x )\).
2. 1. Find the domain of \(\mathrm { fg } ( x )\).
   2. Find the range of \(\mathrm { fg } ( x )\).

1.The function f is given by $$\mathrm { f } ( x ) = x ^ { 2 } - 2 x + 6 , \quad x \geqslant 0$$

1. Find the range of \(f\) . The function \(g\) is given by $$\mathrm { g } ( x ) = 3 + \sqrt { } ( x + 4 ) , \quad x \geqslant 2$$
2. Find \(\operatorname { gf } ( x )\) .
3. Find the domain and range of gf.

1. Express \(4x^2 + 12x + 10\) in the form \((ax + b)^2 + c\), where \(a\), \(b\) and \(c\) are constants. [3]
2. Functions \(f\) and \(g\) are both defined for \(x > 0\). It is given that \(f(x) = x^2 + 1\) and \(fg(x) = 4x^2 + 12x + 10\). Find \(g(x)\). [1]
3. Find \((fg)^{-1}(x)\) and give the domain of \((fg)^{-1}\). [4]

5 Make \(x\) the subject of the equation \(y = \frac { x + 3 } { x - 2 }\).

7 \includegraphics[max width=\textwidth, alt={}, center]{d3c76ceb-cff7-4155-9697-5c302a9d63a9-3_821_688_255_731}

1. The diagram shows part of the curve \(y = 11 - x ^ { 2 }\) and part of the straight line \(y = 5 - x\) meeting at the point \(A ( p , q )\), where \(p\) and \(q\) are positive constants. Find the values of \(p\) and \(q\).
2. The function f is defined for the domain \(x \geqslant 0\) by $$f ( x ) = \begin{cases} 11 - x ^ { 2 } & \text { for } 0 \leqslant x \leqslant p \\ 5 - x & \text { for } x > p \end{cases}$$ Express \(\mathrm { f } ^ { - 1 } ( x )\) in a similar way.