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

8 The curve with equation \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \ln ( 2 x - 3 ) , x > \frac { 3 } { 2 }\), is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-08_630_1173_424_443}

1. The inverse of f is \(\mathrm { f } ^ { - 1 }\).
   1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
   2. State the range of \(\mathrm { f } ^ { - 1 }\).
   3. Sketch, on the axes given on page 9 , the curve with equation \(y = \mathrm { f } ^ { - 1 } ( x )\), indicating the value of the \(y\)-coordinate of the point where the curve intersects the \(y\)-axis.
      (2 marks)
2. The function g is defined by $$\mathrm { g } ( x ) = \mathrm { e } ^ { 2 x } - 4 , \text { for all real values of } x$$
   1. Find \(\mathrm { gf } ( x )\), giving your answer in the form \(( a x - b ) ^ { 2 } - c\), where \(a , b\) and \(c\) are integers.
   2. Write down an expression for \(\mathrm { fg } ( x )\), and hence find the exact solution of the equation \(\operatorname { fg } ( x ) = \ln 5\). \includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-09_1079_1422_233_358}

5 The functions \(f\) and \(g\) are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = x ^ { 2 } - 6 x + 5 , & \text { for } x \geqslant 3 \\ \mathrm {~g} ( x ) = | x - 6 | , & \text { for all real values of } x \end{array}$$

1. Find the range of f.
2. The inverse of f is \(\mathrm { f } ^ { - 1 }\). Find \(\mathrm { f } ^ { - 1 } ( x )\). Give your answer in its simplest form.
3. 1. Find \(\mathrm { gf } ( x )\).
   2. Solve the equation \(\operatorname { gf } ( x ) = 6\).

5 The function f is defined by $$\mathrm { f } ( x ) = 16 x - \mathrm { e } ^ { 2 x } , \text { for all real } x$$ The graph of \(y = \mathrm { f } ( x )\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{bf427498-f1ee-4167-a6f2-ddaa2ff5ef81-12_789_1349_534_347}

1. Find the range of f.
2. The composite function fg is defined by $$\operatorname { fg } ( x ) = \frac { 16 } { x } - \mathrm { e } ^ { \frac { 2 } { x } } , \text { for real } x , x \neq 0$$ Find an expression for \(\operatorname { gg } ( x )\).

1. 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 \(\operatorname { gf } ( x ) = 6\).
   

6. \begin{figure}[h] \end{figure} Figure 1 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 \(\mathrm { f } ^ { - 1 } ( x )\).

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

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.

6. 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 \(y = \mathrm { f } ( x )\).
3. Find an expression for the inverse function, \(\mathrm { f } ^ { - 1 } ( x )\). The function \(g\) is defined by $$\mathrm { g } ( x ) \equiv \mathrm { e } ^ { 2 - x } , \quad x \in \mathbb { R }$$
4. Show that $$\operatorname { fg } ( x ) = x + a - \ln b$$ where \(a\) and \(b\) are integers to be found.

1. The function f is defined by

$$\mathrm { f } ( x ) \equiv 2 + \ln ( 3 x - 2 ) , \quad x \in \mathbb { R } , \quad x > \frac { 2 } { 3 } .$$

1. Find the exact value of \(\mathrm { ff } ( 1 )\).
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
   

6. The function f is defined by $$\mathrm { f } ( x ) \equiv 3 - x ^ { 2 } , \quad x \in \mathbb { R } , \quad x \geq 0 .$$

1. State the range of f.
2. Sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) on the same diagram.
3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain. The function g is defined by $$\mathrm { g } ( x ) \equiv \frac { 8 } { 3 - x } , \quad x \in \mathbb { R } , \quad x \neq 3 .$$
4. Evaluate \(\mathrm { fg } ( - 3 )\).
5. Solve the equation $$\mathrm { f } ^ { - 1 } ( x ) = \mathrm { g } ( x ) .$$

2. 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.

1. Find the range of g in terms of \(a\). Given that \(\operatorname { gf } ( 3 ) = 7\),
2. find the two possible values of \(a\).

7. $$f ( x ) = 1 + \frac { 4 x } { 2 x - 5 } - \frac { 15 } { 2 x ^ { 2 } - 7 x + 5 } , \quad x \in \mathbb { R } , \quad x < 1$$

1. Show that $$f ( x ) = \frac { 3 x + 2 } { x - 1 }$$
2. Find an expression for the inverse function \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.
3. Solve the equation \(\mathrm { f } ( x ) = 2\).

7. \begin{figure}[h] \end{figure} Figure 1 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. \(\quad y = \mathrm { f } ^ { - 1 } ( x )\),
   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.

1. 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 \(\operatorname { gf } ( x ) = \frac { 1 } { 2 }\).
   

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

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. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } ( x ) \equiv 6 x - 1 , \quad x \in \mathbb { R } , \\ & \mathrm {~g} ( x ) \equiv \log _ { 2 } ( 3 x + 1 ) , \quad x \in \mathbb { R } , \quad x > - \frac { 1 } { 3 } \end{aligned}$$

1. Evaluate \(\operatorname { gf } ( 1 )\).
2. Find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).
3. Find, in terms of natural logarithms, the solution of the equation $$\mathrm { fg } ^ { - 1 } ( x ) = 2$$

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

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 \(y = 7\).
   

9 The diagram shows a parabola \(P\) which has equation \(y = \frac { 1 } { 8 } x ^ { 2 }\), and another parabola \(Q\) which is the image of \(P\) under a reflection in the line \(y = x\). The parabolas \(P\) and \(Q\) intersect at the origin and again at a point \(A\).
The line \(L\) is a tangent to both \(P\) and \(Q\). \includegraphics[max width=\textwidth, alt={}, center]{7441c4e6-5448-483b-b100-f8076e7e6cd8-5_1015_1089_623_479}

1. 1. Find the coordinates of the point \(A\).
   2. Write down an equation for \(Q\).
   3. Give a reason why the gradient of \(L\) must be - 1 .
2. 1. Given that the line \(y = - x + c\) intersects the parabola \(P\) at two distinct points, show that $$c > - 2$$
   2. Find the coordinates of the points at which the line \(L\) touches the parabolas \(P\) and \(Q\).
      (No credit will be given for solutions based on differentiation.)

4. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the parabola \(C\) with equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant. The point \(S\) is the focus of \(C\) and the point \(P \left( a p ^ { 2 } \right.\), 2ap) lies on \(C\) where \(p > 0\)

1. Write down the coordinates of \(S\).
2. Write down the length of SP in terms of \(a\) and \(p\). The point \(Q \left( a q ^ { 2 } , 2 a q \right)\), where \(p \neq q\), also lies on \(C\).
   The point \(M\) is the midpoint of \(P Q\).
   Given that \(p q = - 1\)
3. prove that, as \(P\) varies, the locus of \(M\) has equation $$y ^ { 2 } = 2 a ( x - a )$$

9 Answer parts (i) and (iii) on the insert provided. Fig. 9 shows a sketch graph of \(y = \mathrm { f } ( x )\). \begin{figure}[h] \end{figure}

1. On the Insert sketch graphs of
   (A) \(y = 2 \mathrm { f } ( x )\),
   (B) \(y = \mathrm { f } ( - x )\),
   (C) \(y = \mathrm { f } ( x - 2 )\) In each case describe the transformations.
2. Explain why the function \(y = \mathrm { f } ( x )\) does not have an inverse function.
3. The function \(\mathrm { g } ( x )\) is defined as follows: $$\mathrm { g } ( x ) = \mathrm { f } ( x ) \text { for } x \geq 0$$ On the Insert sketch the graph of \(y = \mathrm { g } ^ { - 1 } ( x )\).
4. You are given that \(\mathrm { f } ( x ) = x ^ { 2 } ( x + 2 )\). Calculate the gradient of the curve \(y = \mathrm { f } ( x )\) at the point \(( 1,3 )\).
   Deduce the gradient of the function \(\mathrm { g } ^ { - 1 } ( x )\) at the point where \(x = 3\).
5. Show that \(\mathrm { g } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\) cross where \(x = - 1 + \sqrt { 2 }\). \section*{Insert for question 9.}
6. (A) On the axes below sketch the graph of \(y = 2 \mathrm { f } ( x )\). Describe the transformation. \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-5_563_1102_484_395} Description:
7. (B) On the axes below sketch the graph of \(y = \mathrm { f } ( - x )\). Describe the transformation. \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-5_588_1154_1576_404} Description:
8. (C) On the axes below sketch the graph of \(y = \mathrm { f } ( x - 2 )\). Describe the transformation. \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-6_615_1230_402_406} Description:
9. The function \(\mathrm { g } ( x )\) is defined as follows: $$\mathrm { g } ( x ) = \mathrm { f } ( x ) \text { for } x \geq 0$$ On the axes below sketch the graph of \(y = g ^ { - 1 } ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-6_677_1356_1567_312}

4 \includegraphics[max width=\textwidth, alt={}, center]{28beb431-45d5-4300-88fe-00d05d78790b-05_787_892_267_568} The diagram shows the graph of \(y = \mathrm { f } ( x )\), where $$f ( x ) = \begin{cases} 4 - 4 x , & x \leqslant a , \\ \ln ( b x - 8 ) - 2 , & x \geqslant a . \end{cases}$$ The range of f is \(\mathrm { f } ( x ) \geqslant - 2\).

1. Show that \(a = \frac { 3 } { 2 }\).
2. Find the value of \(b\).
3. Find the exact value of \(\mathrm { ff } ( - 1 )\).
4. Explain why the function f does not have an inverse.

3. The function \(f\) is defined by $$f : x \mapsto \frac { 5 x + 1 } { x ^ { 2 } + x - 2 } - \frac { 3 } { x + 2 } , x > 1$$

1. Show that \(\mathrm { f } ( x ) = \frac { 2 } { x - 1 } , x > 1\).
2. Find \(\mathrm { f } ^ { - 1 } ( x )\). The function g is defined by $$\mathrm { g } : x \mapsto x ^ { 2 } + 5 , \quad x \in \mathbb { R } .$$ (b) Solve \(\mathrm { fg } ( x ) = \frac { 1 } { 4 }\).

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.

9. (i) Find the exact solutions to the equations

1. \(\ln ( 3 x - 7 ) = 5\),
2. \(3 ^ { x } \mathrm { e } ^ { 7 x + 2 } = 15\).
   (ii) The functions f and g are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } + 3 , & x \in \mathbb { R } , \\ \mathrm {~g} ( x ) = \ln ( x - 1 ) , & x \in \mathbb { R } , \quad x > 1 . \end{array}$$
   1. Find \(\mathrm { f } ^ { - 1 }\) and state its domain.
   2. Find fg and state its range.

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

1. State the range of f.
2. 1. Find fg(x).
   2. Solve the equation \(\operatorname { fg } ( x ) = 4\).
3. 1. Explain why the function f does not have an inverse.
   2. The inverse of g is \(\mathrm { g } ^ { - 1 }\). Find \(\mathrm { g } ^ { - 1 } ( x )\).

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

1. State the range of f.
2. 1. Find fg(x).
   2. Solve the equation \(\operatorname { fg } ( x ) = 4\).
3. 1. Explain why the function f does not have an inverse.
   2. The inverse of g is \(\mathrm { g } ^ { - 1 }\). Find \(\mathrm { g } ^ { - 1 } ( x )\).