(a) (i)

The diagram shows the graph of \(y=\mathrm{f}(x)\).
On the same diagram sketch the graph of \(y=\mathrm{f}^{-1}(x)\).

(ii) Describe the relationship between the graph of \(\mathrm{f}(x)\) and the graph of \(\mathrm{f}^{-1}(x)\).

(b) A function g is defined by \(\mathrm{g}(x)=\mathrm{e}^{\sqrt{x-2}}\) for \(x \geqslant 2\).
(i) Find an expression for \(\mathrm{g}^{-1}(x)\).

(ii) Write down the range of \(\mathrm{g}^{-1}\).

(iii) A function h is defined by \(\mathrm{h}(x)=\frac{1}{x^{2}}+2\) for \(x\gt 0\).

Find an expression for \(\operatorname{gh}(x)\) in its simplest form.

It is given that \(\mathrm{f}(x)=2\ln(3x-4)\), for \(x\gt a\), and that \(\mathrm{f}^{-1}\) exists.

(a) Find the least possible value of \(a\).

(b) For your value of \(a\), find the range of \(\mathrm{f}\).

(c) For your value of \(a\), find an expression for \(\mathrm{f}^{-1}(x)\).

(d) It is given that the equation \(\mathrm{f}(x)=\mathrm{f}^{-1}(x)\) has two roots. For your value of \(a\), sketch the graphs of \(y=\mathrm{f}(x)\) and \(y=\mathrm{f}^{-1}(x)\) on the axes. Label each graph. State the intercepts of each graph with the axes. State the equations of any asymptotes.

The function f is defined by \(\mathrm{f}(x)=-2 x^{2}+9 x-10\) for \(0 \leqslant x \leqslant 3\). (a) (i) Write \(\mathrm{f}(x)\) in the form \(a+b(x+c)^{2}\) where \(a, b\) and \(c\) are constants.

(ii) Hence determine whether or not \(\mathrm{f}^{-1}\) exists.

(b) The function g is defined by \(\mathrm{g}(x)=3 \ln (5-2 x)\) for \(0 \leqslant x\lt 2.5\). (i) On the axes, sketch the graph of \(y=\mathrm{g}(x)\).

State the exact values of the intercepts with the coordinate axes and the equation of any asymptote.

(ii) Find an expression for \(\mathrm{g}^{-1}(x)\).

(iii) Find the domain and range of \(\mathrm{g}^{-1}\).

Give each of your answers in exact form.

(a) The function f is defined by \(\mathrm{f}(x)=2 \mathrm{e}^{-x}+3\) for \(x \in \mathbb{R}\). On the axes, sketch the graph of \(y=\mathrm{f}(x)\) and hence, on the same axes, sketch the graph of \(y=\mathrm{f}^{-1}(x)\). Show clearly - the positions of any points where your graphs meet the coordinate axes - the positions of any asymptotes.

(b) The function g is defined by \(\mathrm{g}(x)=2-\frac{3}{\mathrm{e}^{x}+2}\) for \(x \geqslant 0\).

Given that \(\mathrm{g}^{-1}\) exists, find an expression for \(\mathrm{g}^{-1}(x)\) and state its domain.

(a) Write \(2x^2-2x+3\) in the form \(a(x+b)^2+c\), where \(a\), \(b\) and \(c\) are constants.

It is given that \(\mathrm{f}(x)=2x^2-2x+3\), for \(x\leqslant p\).

(b) Write down the greatest value of \(p\) for which \(\mathrm{f}\) has an inverse.

(c) Using this value of \(p\), write down the range of \(\mathrm{f}\).

(d) Using this value of \(p\), find an expression for \(\mathrm{f}^{-1}\).

(a) \(\mathrm{f}(x)=\frac{x}{x-1}\) for \(-10 \leqslant x \leqslant 10, x \neq 1\). The diagram shows the graph of \(y=\mathrm{f}(x)\). (i) Use the diagram to explain why f is a function.

(ii) Find \(\mathrm{ff}(x)\), giving your answer in its simplest form.

(iii) Using your answer to part (ii) state the relationship between the functions f and \(\mathrm{f}^{-1}\).

(iv) Explain how the diagram shows the relationship between f and \(\mathrm{f}^{-1}\).

(b) A function g is defined by \(\mathrm{g}(x)=\frac{x}{x-1}\) for \(x \geqslant 2\). Find the range of g .

(c) A function h is defined by \(\mathrm{h}(x)=\frac{2 x}{3 x+1}\) for the largest possible domain. State the domain of h .

The function f is defined by \(\mathrm{f}(x)=1-4 x-x^{2}\) for all real values of \(x\). (a) Write \(\mathrm{f}(x)\) in the form \(a-(x+b)^{2}\), where \(a\) and \(b\) are constants.

(b) Find the range of f.

The function g is defined by \(\mathrm{g}(x)=1-4 x-x^{2}\) for \(x \geqslant k\), where \(k\) is a constant. (c) State the least possible value of \(k\) such that g has an inverse.

(d) Using your value of \(k\), find \(\mathrm{g}^{-1}(x)\), stating its domain and range.

The functions \(f\) and \(g\) are defined by \(\begin{array}{ll} \mathrm{f}(x)=\frac{3 x^{2}}{4 x-1} & \text { for } x\lt 0 \\ \mathrm{~g}(x)=\frac{1}{x^{2}} & \text { for } x\lt 0 \end{array}\) (a) Explain why the function fg does not exist. (b) Given that the function gf does exist, find and simplify an expression for \(\mathrm{gf}(x)\). (c) Show that \(\mathrm{f}^{-1}(x)\) can be written as \(\frac{p x-\sqrt{x(q x+r)}}{3}\) where \(p, q\) and \(r\) are integers.

The functions \(f\) and \(f g\) are defined by \(\begin{array}{ll} \mathrm{f}(x)=\mathrm{e}^{x^{2}+3} & \text { for } x\lt 0 \\ \mathrm{fg}(x)=\mathrm{e}^{2 x} & \text { for } x\gt \frac{3}{2} \end{array}\) (a) Explain why \(\mathrm{f}^{-1}\) exists. (b) Find an expression for \(\mathrm{f}^{-1}(x)\) and state the domain and range of \(\mathrm{f}^{-1}\). (c) Hence find and simplify an expression for \(\mathrm{g}(x)\).

The functions \(\mathrm f\) and \(\mathrm g\) are defined as follows, for all real values of \(x\).

\(\mathrm f(x)=2x^2-1\)

\(\mathrm g(x)=\mathrm e^x+1\)

(a) Solve the equation \(\mathrm{fg}(x)=8\).

(b) For each of the functions \(\mathrm f\) and \(\mathrm g\), either explain why the inverse function does not exist or find the inverse function, stating its domain.

The function \(f\) is defined by

\(f(x)=\frac{\sqrt{4x^2-1}}{2x}\quad\text{for }0.5\leqslant x\leqslant1.5.\)

The diagram shows a sketch of \(y=f(x)\).

(a)

(i) It is given that \(f^{-1}\) exists. Find the domain and range of \(f^{-1}\).

(ii) Find an expression for \(f^{-1}(x)\).

(b) The function \(g\) is defined by \(g(x)=e^{x^2}\) for all real \(x\). Show that \(gf(x)=e^{\left(1-\frac{a}{bx^2}\right)}\), where \(a\) and \(b\) are integers.

The functions \(\mathrm f\) and \(\mathrm g\) are defined, for \(x\gt 0\), by

\(\mathrm f(x)=\frac{2x^2-1}{3x}, \qquad \mathrm g(x)=\frac1x.\)

(a) Find and simplify an expression for \(\mathrm{fg}(x)\).

(b)

(i) Given that \(\mathrm f^{-1}\) exists, write down the range of \(\mathrm f^{-1}\).

(ii) Show that

\(\mathrm f^{-1}(x)=\frac{px+\sqrt{qx^2+r}}{4},\)

where \(p\), \(q\) and \(r\) are integers.

(a) The function \(\mathrm f\) is defined, for all real \(x\), by

\(\mathrm f(x)=13-4x-2x^2.\)

(i) Write \(\mathrm f(x)\) in the form \(a+b(x+c)^2\), where \(a\), \(b\) and \(c\) are constants.

(ii) Hence write down the range of \(\mathrm f\).

(b) The function \(\mathrm g\) is defined, for \(x\geq1\), by

\(\mathrm g(x)=\sqrt{x^2+2x-1}.\)

(i) Given that \(\mathrm g^{-1}(x)\) exists, write down the domain and range of \(\mathrm g^{-1}\).

(ii) Show that

\(\mathrm g^{-1}(x)=-1+\sqrt{px^2+q},\)

where \(p\) and \(q\) are integers.

(a) The function \(g\) is defined by

\(g(x)=3+\frac1x,\qquad x\geq1.\)

Find an expression for \(g^{-1}(x)\), and state the domain and range of \(g^{-1}\).

(b) The function \(h\) is defined by

\(h(x)=2\ln(3x-1),\qquad x\gt \frac23.\)

The graph of \(y=h(x)\) intersects the line \(y=x\) at two distinct points. On the same axes, sketch the graphs of \(y=h(x)\) and \(y=h^{-1}(x)\).

The function \(f\) is defined by

\(f:x\mapsto(2x+3)^2,\qquad x\gt0.\)

(a) State the range of \(f\).

(b) Explain why \(f\) has an inverse.

(c) Find \(f^{-1}\).

(d) State the domain of \(f^{-1}\).

(e) Given that

\(g:x\mapsto\ln(x+4),\qquad x\gt0,\)

find the exact solution of \(fg(x)=49\).

(a) It is given that \(g(x)=6x^4+5\) for all real \(x\).

(i) Explain why \(g\) is a function but does not have an inverse.

(ii) Find \(g^2(x)\) and state its domain.

It is given that \(h(x)=6x^4+5\) for \(x\leq k\).

(iii) State the greatest value of \(k\) such that \(h^{-1}\) exists.

(iv) For this value of \(k\), find \(h^{-1}(x)\).

(b) The function \(p\) is defined by \(p(x)=3e^x+2\) for all real \(x\).

(i) State the range of \(p\).

(ii) Sketch and label the graphs of \(y=p(x)\) and \(y=p^{-1}(x)\). State the coordinates of any points of intersection with the coordinate axes.

(iii) Hence explain why the equation \(p(x)=p^{-1}(x)\) has no solutions.

\(f:x\mapsto e^{3x}\) for \(x\in\mathbb R\), and \(g:x\mapsto2x^2+1\) for \(x\geq0\).

(i) Write down the range of \(g\).

(ii) Show that \(f^{-1}g(\sqrt{62})=\ln5\).

(iii) Solve \(f'(x)=6g''(x)\), giving your answer in the form \(\ln a\), where \(a\) is an integer.

(iv) Sketch the graph of \(y=g\) and the graph of \(y=g^{-1}\), showing the points where the graphs meet the coordinate axes.

The functions \(f\) and \(g\) are defined by \(f(x)=5x-2\) for \(x\gt 1\), and \(g(x)=4x^2-9\) for \(x\gt 0\).

(a) (i) State the range of \(g\).

(ii) Find the domain of \(gf\).

(iii) Showing all your working, find the exact solutions of \(gf(x)=45\).

(b) The function \(h\) is defined by \(h(x)=\sqrt{x^2-1}\) for \(x\leq-1\).

(i) State the geometrical relationship between the graphs of \(y=h(x)\) and \(y=h^{-1}(x)\).

(ii) Find an expression for \(h^{-1}(x)\).

(a) The function \(f\) is defined by \(f(x)=\sqrt{1+x^2}\), for all real values of \(x\). The graph of \(y=f(x)\) is given.

(i) Explain, with reference to the graph, why \(f\) does not have an inverse.

(ii) Find \(f^2(x)\).

(b) The function \(g\) is defined, for \(x\gt k\), by \(g(x)=\sqrt{1+x^2}\), and \(g\) has an inverse.

(i) Write down a possible value for \(k\).

(ii) Find \(g^{-1}(x)\).

(c) The function \(h\) is defined, for all real values of \(x\), by \(h(x)=4e^x+2\). Sketch the graph of \(y=h(x)\). Hence, on the same axes, sketch the graph of \(y=h^{-1}(x)\). Give the coordinates of any points where your graphs meet the coordinate axes.

The function \(\mathrm f\) is defined by

\(\mathrm f(x)=\frac1{2x-5},\qquad x\gt2.5.\)

(i) Find an expression for \(\mathrm f^{-1}(x)\).

(ii) State the domain of \(\mathrm f^{-1}(x)\).

(iii) Find an expression for \(\mathrm f^2(x)\), giving your answer in the form \(\dfrac{ax+b}{cx+d}\), where \(a\), \(b\), \(c\) and \(d\) are integers to be found.

The function \(\mathrm f\) is defined by

\(\mathrm f(x)=\frac1{2x-5},\qquad x\gt2.5.\)

(i) Find an expression for \(\mathrm f^{-1}(x)\).

(ii) State the domain of \(\mathrm f^{-1}(x)\).

(iii) Find an expression for \(\mathrm f^2(x)\), giving your answer in the form \(\dfrac{ax+b}{cx+d}\), where \(a\), \(b\), \(c\) and \(d\) are integers to be found.

(a) \(f(x)=3-\cos2x\), for \(0\leq x\leq\dfrac{\pi}{2}\).

(i) Write down the range of \(f\).

(ii) Find the exact value of \(f^{-1}(2.5)\).

(b) \(g(x)=3-x^2\), for \(x\in\mathbb R\). Find the exact solutions of \(g^2(x)=-6\).

\(f(x)=5+\sin\frac{x}{4} \quad\text{for}\quad 0\leq x\leq2\pi\text{ radians}\)

\(g(x)=x-\frac{\pi}{3} \quad\text{for}\quad x\in\mathbb R\)

(i) Write down the range of \(f(x)\).

(ii) Find \(f^{-1}(x)\) and write down its range.

(iii) Solve \(2fg(x)=11\).

(a) It is given that \(f(x)=3e^{-4x}+5\), for \(x\in\mathbb R\).

(i) State the range of \(f\).

(ii) Find \(f^{-1}\) and state its domain.

(b) It is given that \(g(x)=x^2+5\) and \(h(x)=\ln x\), for \(x\gt 0\). Solve \(hg(x)=2\).

(a) Functions \(f\) and \(g\) are such that, for \(x\in\mathbb R\), \(f(x)=x^2+3\) and \(g(x)=4x-1\).

(i) State the range of \(f\).

(ii) Solve \(fg(x)=4\).

(b) A function \(h\) is such that \(h(x)=\dfrac{2x+1}{x-4}\), for \(x\in\mathbb R\), \(x\neq4\).

(i) Find \(h^{-1}(x)\) and state its range.

(ii) Find \(h^2(x)\), giving your answer in its simplest form.

Functions \(f\) and \(g\) are defined, for \(x\gt 0\), by \(f(x)=\ln x\) and \(g(x)=2x^2+3\).

(i) Write down the range of \(f\).

(ii) Write down the range of \(g\).

(iii) Find the exact value of \(f^{-1}g(4)\).

(iv) Find \(g^{-1}(x)\) and state its domain.

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

\(f(x)=(x+2)^2+1,\)

\(g(x)=\dfrac{x-2}{2x-1}\), where \(x\ne\dfrac12\).

(i) Find \(f^2(-3)\).

(ii) Show that \(g^{-1}(x)=g(x)\).

(iii) Solve \(gf(x)=\dfrac8{19}\).