It is given that \(\mathrm{f}(x)=\ln(2x+5)\) for \(x\gt a\), where \(a\) is a constant.

(a) Write down the least possible value of \(a\).

(b) Using your value of \(a\), write down the range of \(\mathrm{f}\).

It is also given that \(\mathrm{g}(x)=x^2+1\), for \(x\in\mathbb{R}\).

(c) Using your value of \(a\), solve the equation \(\mathrm{fg}(x)=4\). Give your answers in exact form.

The diagrams show four different relations.

(b) State whether each relation is one-one, many-one, and whether it is its own inverse.

(c) The functions are given by \(\mathrm{f}:x\mapsto\sin x\), for \(30^\circ\leqslant x\leqslant a^\circ\), and \(\mathrm{g}:x\mapsto\sqrt{x-\frac12}\), for \(x\geqslant\frac12\). Given that \(\mathrm{g}\mathrm{f}\) exists, find the largest possible value of \(a\), the range of \(\mathrm{g}\mathrm{f}\), and explain why \(\mathrm{g}^2\) does not exist.

Functions \(f\) and \(g\) are such that

\(\mathrm{f}(x)=\frac{3x}{x+4}\) for \(x\gt 0\).

\(\mathrm{g}(x)=\sqrt{x+2}\) for \(x\gt -2\).

Solve the equation \(\mathrm{fg}(x)=1\).

It is given that \(\mathrm f(x)=2\mathrm e^x+a\) for \(x\geqslant0\), where \(a\) is an integer, and \(\mathrm g(x)=\sqrt{x-1}\) for \(x\geqslant1\).

(a) Find the least value of \(a\) so that the function \(\mathrm{gf}\) exists for all \(x\geqslant0\).

(b) In the case where \(a=5\), solve the equation \(\mathrm{gf}(x)=3\). Give your answer correct to 3 decimal places.

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

\(\mathrm f:x\mapsto2\sin x+3\cos x\)

\(\mathrm g:x\mapsto \mathrm e^{3x}-1\)

(a) Find the value of \(\mathrm{fg}(0)\).

(b) Find \(\mathrm{gg}(x)\) in terms of \(x\), giving your answer in its simplest form.

(c) Solve the equation \(\mathrm g^{-1}(x)=\frac13\ln5\).

(a) It is given that \(\mathrm f:x\mapsto2x^2\), for \(x\ge0\), and \(\mathrm g:x\mapsto2x+1\), for \(x\ge0\).

Each of the expressions in the table can be written as one of \(\mathrm f'\), \(\mathrm f''\), \(\mathrm g'\), \(\mathrm g''\), \(\mathrm{fg}\), \(\mathrm{gf}\), \(\mathrm f^2\), \(\mathrm g^2\), \(\mathrm f^{-1}\), \(\mathrm g^{-1}\). Complete the table.

(b) It is given that \(\mathrm h(x)=(x-1)^2+3\), for \(x\ge a\). The value of \(a\) is as small as possible such that \(\mathrm h^{-1}\) exists.

(i) Write down the value of \(a\).

(ii) Write down the range of \(\mathrm h\).

(iii) Find \(\mathrm h^{-1}(x)\) and state its domain.

The functions \(f(x)\) and \(g(x)\) are defined as follows for \(x\gt -\frac23\) by

\(f(x)=x^2+1, \qquad g(x)=\ln(3x+2).\)

(a) Find \(fg(x)\).

(b) Solve the equation \(fg(x)=5\), giving your answer in exact form.

(c) Solve the equation \(gg(x)=1\).

The functions \(\mathrm f\) and \(\mathrm g\) are defined as follows.

\(\mathrm f(x)=x^2+4x\quad\text{for }x\in\mathbb R,\)

\(\mathrm g(x)=1+\mathrm e^{2x}\quad\text{for }x\in\mathbb R.\)

(a) Find the range of \(\mathrm f\).

(b) Write down the range of \(\mathrm g\).

(c) Find the exact solution of the equation \(\mathrm{fg}(x)=21\), giving your answer as a single logarithm.

The functions \(\mathrm f\) and \(\mathrm g\) are defined for \(x\gt 1\) by

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

(a) Find \(\mathrm{fg}(x)\).

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

(c) Without using a calculator, solve the equation \(\mathrm f(x)=\mathrm g(x)\).

\(f(x)=3+e^x\quad\text{for }x\in\mathbb R\)

\(g(x)=9x-5\quad\text{for }x\in\mathbb R\)

(a) Find the range of \(f\) and of \(g\).

(b) Find the exact solution of \(f^{-1}(x)=g'(x)\).

(c) Find the solution of \(g^2(x)=112\).

(a) It is given that

\(f:x\mapsto\sqrt{x}\quad\text{for }x\geq0,\)

\(g:x\mapsto x+5\quad\text{for }x\geq0.\)

Identify each of the following functions with one of \(f^{-1}\), \(g^{-1}\), \(fg\), \(gf\), \(f^2\), \(g^2\).

(i) \(\sqrt{x+5}\)

(ii) \(x-5\)

(iii) \(x^2\)

(iv) \(x+10\)

(b) It is given that

\(h(x)=a+\frac{b}{x^2},\)

where \(a\) and \(b\) are constants.

(i) Why is \(-2\leq x\leq2\) not a suitable domain for \(h(x)\)?

(ii) Given that \(h(1)=4\) and \(h'(1)=16\), find the value of \(a\) and of \(b\).

The functions \(f\) and \(g\) are defined for real values of \(x\gt 1\) by

\(f(x)=4x-3, \qquad g(x)=\frac{2x+1}{3x-1}.\)

(i) Find \(gf(x)\).

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

(iii) Solve \(fg(x)=x-1\).

The function \(g\) is defined, for \(x\gt -\dfrac12\), by

\(g(x)=\frac{3}{2x+1}.\)

(i) Show that \(g'(x)\) is always negative.

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

The function \(h\) is defined, for all real \(x\), by \(h(x)=kx+3\), where \(k\) is a constant.

(iii) Find an expression for \(hg(x)\).

(iv) Given that \(hg(0)=5\), find the value of \(k\).

(v) State the domain of \(hg\).