$\newcommand{\D}{\displaystyle}$
1 (CIE 2012, s, paper 12, question 10)
(a) It is given that $\D f(x) =\frac{1}{2+x}$ for $\D x \not= -2, x\in R.$
(i) Find $\D f ″(x).$ [2]
(ii) Find $\D f^{-1} (x).$ [2]
(iii) Solve $\D f^2(x) = -1.$ [3]
(b) The functions g, h and k are defined, for $\D x\in R,$ by
\begin{eqnarray*}
g(x)&=&\frac{1}{x+5},x\not=-5\\
h(x)&=&x^2-1,\\
k(x)&=&2x+1.
\end{eqnarray*}
Express the following in terms of g, h and/or k.
(i) $\D \frac{1}{(x^2-1)+5}$ [1]
(ii) $\D \frac{2}{x+5}+1$ [1]
2 (CIE 2012, s, paper 21, question 12or)
A function g is defined by $\D g : x \mapsto 5x^2 + px + 72,$ where $\D p$ is a constant. The function can also be written as $\D g : x \mapsto 5(x - 4)^2 + q.$
(i) Find the value of $\D p$ and of $\D q.$ [3]
(ii) Find the range of the function g. [1]
(iii) Sketch the graph of the function on the axes provided. [2]
(iv) Given that the function $\D h$ is defined by $\D h : x \mapsto \ln x,$ where $\D x > 0,$ solve the equation $\D gh(x) = 12.$ [4]
3 (CIE 2012, w, paper 11, question 9)
A function g is such that $\D g(x) = \frac{1}{2x-1}$ for $\D 1 \le x \le 3.$
(i) Find the range of $\D g.$ [1]
(ii) Find $\D g^{-1}(x).$ [2]
(iii) Write down the domain of $\D g^{-1}(x).$ [1]
(iv) Solve $\D g^2(x) = 3.$ [3]
4 (CIE 2012, w, paper 23, question 12either)
(i) Express $\D 4x^2 + 32x + 55$ in the form $\D (ax + b)^2 + c,$ where a, b and c are constants and a is positive. [3]
The functions f and g are defined by
\begin{eqnarray*}
f:x&\mapsto& 4x^2+32x+55 \mbox{ for } x>-4\\
g:x&\mapsto&\frac{1}{x}\mbox{ for }x>0.
\end{eqnarray*}
(ii) Find $\D f^{-1}(x).$ [3]
(iii) Solve the equation $\D fg(x) = 135.$ [4]
5 (CIE 2012, w, paper 23, question 12or)
The functions h and k are defined by
\begin{eqnarray*}
h:x&\mapsto& \sqrt{2x-7} \mbox{ for } x>c\\
k:x&\mapsto&\frac{3x-4}{x-2}\mbox{ for }x>2.
\end{eqnarray*}
(i) State the least possible value of c. [1]
(ii) Find $\D h^{-1}(x).$ [2]
(iii) Solve the equation $\D k(x) = x.$ [3]
(iv) Find an expression for the function $\D k^2,$ in the form $\D k^2 : x \mapsto a + \frac{b}{x}$ where a and b are constants. [4]
6 (CIE 2013, s, paper 21, question 11)
A one-one function f is defined by $\D f(x)= (x- 1)^2- 5 $ for $\D x \ge k .$
(i) State the least value that k can take. [1]
For this least value of k
(ii) write down the range of f, [1]
(iii) find $\D f^{-1}(x),$ [2]
(iv) sketch and label, on the axes below, the graph of $\D y = f(x)$ and of $\D y= f^{-1}(x),$ [2]
(v) find the value of x for which $\D f(x)= f^{-1}(x).$ [2]
7 (CIE 2013, w, paper 11, question 12)
(a) A function f is such that $\D f (x)= 3x^2- 1$ for $\D - 10 \le x \le 8.$
(i) Find the range of f. [3]
(ii) Write down a suitable domain for f for which $f^{-1}$ exists. [1]
(b) Functions g and h are defined by $\D g(x)= 4e^x- 2$ for $\D x \in R,$ $h(x) = \ln 5x$ for $\D x > 0.$
(i) Find $\D g^{-1} (x).$ [2]
(ii) Solve $\D gh(x) = 18.$ [3]
8 (CIE 2013, w, paper 13, question 5)
For $\D x\in R,$ the functions f and g are defined by
\begin{eqnarray*}
f(x)&=&2x^3,\\
g(x)&=&4x-5x^2.
\end{eqnarray*}
(i) Express $\D f^2\left(\frac{1}{2}\right)$ as a power of 2. [2]
(ii) Find the values of x for which f and g are increasing at the same rate with respect to x. [4]
9 (CIE 2014, s, paper 21, question 12)
The functions f and g are defined by
\begin{eqnarray*}
f(x)&=&\frac{2x}{x+1}\mbox{ for } x>0,\\
g(x)&=&\sqrt{x+1}\mbox{ for } x>-1.
\end{eqnarray*}
(i) Find $\D fg(8)$. [2]
(ii) Find an expression for $\D f^2(x),$ giving your answer in the form $\D \frac{ax}{bx+c},$ where a, b and c are integers to be found. [3]
(iii) Find an expression for $\D g^{-1}(x),$ stating its domain and range. [4]
(iv) On the same axes, sketch the graphs of $\D y=g(x)$ and $\D y=g^{-1}(x),$ indicating the geometrical relationship between the graphs. [3]
10 (CIE 2014, s, paper 22, question 11)
The functions f and g are defined, for real values of x greater than 2, by
\begin{eqnarray*}
f(x)&=&2^x-1,\\
g(x)&=&x(x+1).
\end{eqnarray*}
(i) State the range of f. [1]
(ii) Find an expression for $\D f^{-1} (x),$ stating its domain and range. [4]
(iii) Find an expression for $\D gf (x)$ and explain why the equation $\D gf (x) = 0$ has no solutions. [4]
11 (CIE 2014, s, paper 23, question 12)
The function f is such that $\D f(x) = \sqrt{x-3}$ for $\D 4\le x\le 28.$
(i) Find the range of f. [2]
(ii) Find $\D f^2 (12).$ [2]
(iii) Find an expression for $\D f^{-1} (x).$ [2]
The function g is defined by $\D g(x)=\frac{120}{x}$ for $\D x\ge 0.$
(iv) Find the value of x for which $\D gf (x) = 20.$ [3]
12 (CIE 2014, w, paper 21, question 4)
The functions f and g are defined for real values of x by
\begin{eqnarray*}
f(x)&=&\sqrt{x-1}-3 \mbox{ for } x>1,\\
g(x)&=& \frac{x-2}{2x-3} \mbox{ for }x>2.
\end{eqnarray*}
(i) Find $\D gf(37).$ [2]
(ii) Find an expression for $\D f^{-1} (x).$ [2]
(iii) Find an expression for $\D g^{-1} (x) .$ [2]
13 (CIE 2014, w, paper 23, question 7)
The functions f and g are defined for real values of x by
\begin{eqnarray*}
f(x)&=& \frac{2}{x}+1 \mbox{ for }x>1,\\
g(x)&=&x^2+2.
\end{eqnarray*}
Find an expression for
(i) $\D f^{-1}(x),$ [2]
(ii) $\D gf(x),$ [2]
(iii) $\D fg(x).$ [2]
(iv) Show that $\D ff(x)=\frac{3x+2}{x+2}$ and solve $\D ff(x)=x.$ [4]
14 (CIE 2015, s, paper 11, question 8)
It is given that
\begin{eqnarray*}
f(x)&=&3e^{2x} \mbox{ for }x\ge 0,\\
g(x)&=&(x+2)^2+5 \mbox{ for } x\ge 0.
\end{eqnarray*}
(i) Write down the range of f and of g. [2]
(ii) Find $\D g^{-1},$ stating its domain. [3]
(iii) Find the exact solution of $\D gf(x) = 41.$ [4]
(iv) Evaluate $\D f'(\ln 4).$ [2]
Answers
1.(a)(i) $\D 2(2+x)^{-3}$
(ii) $\D \frac{1-2x}{x}$
(iii) $\D x=-\frac{7}{3}$
(b) $\D gh,kg$
2. (i) $p=-40,q=-8$
(ii) $g(x)>-8$
(iii)
(iv) $\D x=e^2,x=e^6$
3.(i) $\D 0.2\le x\le 1$
(ii) $\D g^{-1}(x)=\frac{1+x}{2x}$
(iii) $\D 0.2\le x\le 1$
(iv) $x+1.25$
4(i) $\D (2x+8)^2-9$
(ii) $\D f^{-1}=\frac{\sqrt{x+9}-8}{2}$
(iii) $\D x=0.5$
5(i) 3.5 (ii) $\D h^{-1}(x)=\frac{x^2+7}{2}$
(iii) $\D x=4$ (iv) 5-4/x
6(i)1 (ii) $\D f\ge -5$
(iii) $\D 1+\sqrt{x+5}$ (v)4
7(a)(i) $\D -1\le y\le 299$
(ii) $\D x\ge 0$
(b)(i) $\D \ln\left(\frac{x+2}{4}\right)$
(ii) $\D x=1$
8(i) $\D 2^{-5}$ (ii) $\D x=1/3,-2$
9(i) $\D 3/2 $
(ii) $\D 4x/(3x+1)$
(iii) $\D g^{-1}(x)=x^2-1$
D: $x>0$ R:$\D g^{-1}(x)>-1$
(iv)
10(i) $\D f(x)>3$
(ii) $\D f^{-1}(x)=\log_2(x+1)$
$x>3,y>2$
(iii) no solution
11(i) $\D 3<f<7$
(ii) $\D 2+\sqrt{2}$
(iii) $\D f^{-1}(x)=(x-2)^2+3$
(iv) $\D x=19$
12(i) $\D 1/3$
(ii) $\D (x+3)^2+1$
(iii)$\D \frac{3x-2}{2x-1}$
13(i) $\D 2/(x-1)$
(ii) $\D gf(x)=(2/x+1)^2+2$
(iii)$\D fg(x)=2/(x^2+2)+1$
(iv) $\D x=2$
1 (CIE 2012, s, paper 12, question 10)
(a) It is given that $\D f(x) =\frac{1}{2+x}$ for $\D x \not= -2, x\in R.$
(i) Find $\D f ″(x).$ [2]
(ii) Find $\D f^{-1} (x).$ [2]
(iii) Solve $\D f^2(x) = -1.$ [3]
(b) The functions g, h and k are defined, for $\D x\in R,$ by
\begin{eqnarray*}
g(x)&=&\frac{1}{x+5},x\not=-5\\
h(x)&=&x^2-1,\\
k(x)&=&2x+1.
\end{eqnarray*}
Express the following in terms of g, h and/or k.
(i) $\D \frac{1}{(x^2-1)+5}$ [1]
(ii) $\D \frac{2}{x+5}+1$ [1]
2 (CIE 2012, s, paper 21, question 12or)
A function g is defined by $\D g : x \mapsto 5x^2 + px + 72,$ where $\D p$ is a constant. The function can also be written as $\D g : x \mapsto 5(x - 4)^2 + q.$
(i) Find the value of $\D p$ and of $\D q.$ [3]
(ii) Find the range of the function g. [1]
(iii) Sketch the graph of the function on the axes provided. [2]
(iv) Given that the function $\D h$ is defined by $\D h : x \mapsto \ln x,$ where $\D x > 0,$ solve the equation $\D gh(x) = 12.$ [4]
3 (CIE 2012, w, paper 11, question 9)
A function g is such that $\D g(x) = \frac{1}{2x-1}$ for $\D 1 \le x \le 3.$
(i) Find the range of $\D g.$ [1]
(ii) Find $\D g^{-1}(x).$ [2]
(iii) Write down the domain of $\D g^{-1}(x).$ [1]
(iv) Solve $\D g^2(x) = 3.$ [3]
4 (CIE 2012, w, paper 23, question 12either)
(i) Express $\D 4x^2 + 32x + 55$ in the form $\D (ax + b)^2 + c,$ where a, b and c are constants and a is positive. [3]
The functions f and g are defined by
\begin{eqnarray*}
f:x&\mapsto& 4x^2+32x+55 \mbox{ for } x>-4\\
g:x&\mapsto&\frac{1}{x}\mbox{ for }x>0.
\end{eqnarray*}
(ii) Find $\D f^{-1}(x).$ [3]
(iii) Solve the equation $\D fg(x) = 135.$ [4]
5 (CIE 2012, w, paper 23, question 12or)
The functions h and k are defined by
\begin{eqnarray*}
h:x&\mapsto& \sqrt{2x-7} \mbox{ for } x>c\\
k:x&\mapsto&\frac{3x-4}{x-2}\mbox{ for }x>2.
\end{eqnarray*}
(i) State the least possible value of c. [1]
(ii) Find $\D h^{-1}(x).$ [2]
(iii) Solve the equation $\D k(x) = x.$ [3]
(iv) Find an expression for the function $\D k^2,$ in the form $\D k^2 : x \mapsto a + \frac{b}{x}$ where a and b are constants. [4]
6 (CIE 2013, s, paper 21, question 11)
A one-one function f is defined by $\D f(x)= (x- 1)^2- 5 $ for $\D x \ge k .$
(i) State the least value that k can take. [1]
For this least value of k
(ii) write down the range of f, [1]
(iii) find $\D f^{-1}(x),$ [2]
(iv) sketch and label, on the axes below, the graph of $\D y = f(x)$ and of $\D y= f^{-1}(x),$ [2]
(v) find the value of x for which $\D f(x)= f^{-1}(x).$ [2]
7 (CIE 2013, w, paper 11, question 12)
(a) A function f is such that $\D f (x)= 3x^2- 1$ for $\D - 10 \le x \le 8.$
(i) Find the range of f. [3]
(ii) Write down a suitable domain for f for which $f^{-1}$ exists. [1]
(b) Functions g and h are defined by $\D g(x)= 4e^x- 2$ for $\D x \in R,$ $h(x) = \ln 5x$ for $\D x > 0.$
(i) Find $\D g^{-1} (x).$ [2]
(ii) Solve $\D gh(x) = 18.$ [3]
8 (CIE 2013, w, paper 13, question 5)
For $\D x\in R,$ the functions f and g are defined by
\begin{eqnarray*}
f(x)&=&2x^3,\\
g(x)&=&4x-5x^2.
\end{eqnarray*}
(i) Express $\D f^2\left(\frac{1}{2}\right)$ as a power of 2. [2]
(ii) Find the values of x for which f and g are increasing at the same rate with respect to x. [4]
9 (CIE 2014, s, paper 21, question 12)
The functions f and g are defined by
\begin{eqnarray*}
f(x)&=&\frac{2x}{x+1}\mbox{ for } x>0,\\
g(x)&=&\sqrt{x+1}\mbox{ for } x>-1.
\end{eqnarray*}
(i) Find $\D fg(8)$. [2]
(ii) Find an expression for $\D f^2(x),$ giving your answer in the form $\D \frac{ax}{bx+c},$ where a, b and c are integers to be found. [3]
(iii) Find an expression for $\D g^{-1}(x),$ stating its domain and range. [4]
(iv) On the same axes, sketch the graphs of $\D y=g(x)$ and $\D y=g^{-1}(x),$ indicating the geometrical relationship between the graphs. [3]
10 (CIE 2014, s, paper 22, question 11)
The functions f and g are defined, for real values of x greater than 2, by
\begin{eqnarray*}
f(x)&=&2^x-1,\\
g(x)&=&x(x+1).
\end{eqnarray*}
(i) State the range of f. [1]
(ii) Find an expression for $\D f^{-1} (x),$ stating its domain and range. [4]
(iii) Find an expression for $\D gf (x)$ and explain why the equation $\D gf (x) = 0$ has no solutions. [4]
11 (CIE 2014, s, paper 23, question 12)
The function f is such that $\D f(x) = \sqrt{x-3}$ for $\D 4\le x\le 28.$
(i) Find the range of f. [2]
(ii) Find $\D f^2 (12).$ [2]
(iii) Find an expression for $\D f^{-1} (x).$ [2]
The function g is defined by $\D g(x)=\frac{120}{x}$ for $\D x\ge 0.$
(iv) Find the value of x for which $\D gf (x) = 20.$ [3]
12 (CIE 2014, w, paper 21, question 4)
The functions f and g are defined for real values of x by
\begin{eqnarray*}
f(x)&=&\sqrt{x-1}-3 \mbox{ for } x>1,\\
g(x)&=& \frac{x-2}{2x-3} \mbox{ for }x>2.
\end{eqnarray*}
(i) Find $\D gf(37).$ [2]
(ii) Find an expression for $\D f^{-1} (x).$ [2]
(iii) Find an expression for $\D g^{-1} (x) .$ [2]
13 (CIE 2014, w, paper 23, question 7)
The functions f and g are defined for real values of x by
\begin{eqnarray*}
f(x)&=& \frac{2}{x}+1 \mbox{ for }x>1,\\
g(x)&=&x^2+2.
\end{eqnarray*}
Find an expression for
(i) $\D f^{-1}(x),$ [2]
(ii) $\D gf(x),$ [2]
(iii) $\D fg(x).$ [2]
(iv) Show that $\D ff(x)=\frac{3x+2}{x+2}$ and solve $\D ff(x)=x.$ [4]
14 (CIE 2015, s, paper 11, question 8)
It is given that
\begin{eqnarray*}
f(x)&=&3e^{2x} \mbox{ for }x\ge 0,\\
g(x)&=&(x+2)^2+5 \mbox{ for } x\ge 0.
\end{eqnarray*}
(i) Write down the range of f and of g. [2]
(ii) Find $\D g^{-1},$ stating its domain. [3]
(iii) Find the exact solution of $\D gf(x) = 41.$ [4]
(iv) Evaluate $\D f'(\ln 4).$ [2]
Answers
1.(a)(i) $\D 2(2+x)^{-3}$
(ii) $\D \frac{1-2x}{x}$
(iii) $\D x=-\frac{7}{3}$
(b) $\D gh,kg$
2. (i) $p=-40,q=-8$
(ii) $g(x)>-8$
(iii)
(iv) $\D x=e^2,x=e^6$
3.(i) $\D 0.2\le x\le 1$
(ii) $\D g^{-1}(x)=\frac{1+x}{2x}$
(iii) $\D 0.2\le x\le 1$
(iv) $x+1.25$
4(i) $\D (2x+8)^2-9$
(ii) $\D f^{-1}=\frac{\sqrt{x+9}-8}{2}$
(iii) $\D x=0.5$
5(i) 3.5 (ii) $\D h^{-1}(x)=\frac{x^2+7}{2}$
(iii) $\D x=4$ (iv) 5-4/x
6(i)1 (ii) $\D f\ge -5$
(iii) $\D 1+\sqrt{x+5}$ (v)4
7(a)(i) $\D -1\le y\le 299$
(ii) $\D x\ge 0$
(b)(i) $\D \ln\left(\frac{x+2}{4}\right)$
(ii) $\D x=1$
8(i) $\D 2^{-5}$ (ii) $\D x=1/3,-2$
9(i) $\D 3/2 $
(ii) $\D 4x/(3x+1)$
(iii) $\D g^{-1}(x)=x^2-1$
D: $x>0$ R:$\D g^{-1}(x)>-1$
(iv)
10(i) $\D f(x)>3$
(ii) $\D f^{-1}(x)=\log_2(x+1)$
$x>3,y>2$
(iii) no solution
11(i) $\D 3<f<7$
(ii) $\D 2+\sqrt{2}$
(iii) $\D f^{-1}(x)=(x-2)^2+3$
(iv) $\D x=19$
12(i) $\D 1/3$
(ii) $\D (x+3)^2+1$
(iii)$\D \frac{3x-2}{2x-1}$
13(i) $\D 2/(x-1)$
(ii) $\D gf(x)=(2/x+1)^2+2$
(iii)$\D fg(x)=2/(x^2+2)+1$
(iv) $\D x=2$
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