(a) Show that \(2x^2+5x+3\) can be written in the form \(2(x+a)^2+b\), where \(a\) and \(b\) are constants to be found.

(b) Hence write down the coordinates of the stationary point on the curve \(y=2x^2+5x+3\).

A function \(\mathrm{f}\) is such that \(\mathrm{f}(x)=2x^2+5x+3\), for \(x\geqslant p\), where \(p\) is a constant. It is given that \(\mathrm{f}^{-1}\) exists.

(c)(i) Write down the least possible value of \(p\).

(ii) Using your value of \(p\), sketch the graphs of \(y=\mathrm{f}(x)\) and \(y=\mathrm{f}^{-1}(x)\). Label each graph. State the intercepts of each of the graphs with the axes.

A function f is such that \(\mathrm{f}(x)=2+\mathrm{e}^{-3 x}, \quad x \in \mathbb{R}\). (a) Write down the range of f .

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

(c) On the axes, sketch the graphs of \(y=\mathrm{f}(x)\) and \(y=\mathrm{f}^{-1}(x)\), stating the coordinates of the points where the curves meet the coordinate axes. State the equations of any asymptotes. Label your curves.

A function g is such that \(\mathrm{g}(x)=x^{\frac{3}{2}}+4, \quad x \geqslant 0\). (d) Find the exact solution of the equation \(\mathrm{gf}(x)=12\).

(a) The diagram shows the graph of \(y=f(x)\), where \(f\) is defined by

\(f(x)=\frac{3x}{\sqrt{5x+1}}\quad\text{for }0\leq x\leq3.\)

(i) Given that \(f\) is a one-one function, find the domain and range of \(f^{-1}\).

(ii) Solve the equation \(f(x)=x\).

(iii) Sketch the graph of \(y=f^{-1}(x)\).

(b) The functions \(g\) and \(h\) are defined by

\(g(x)=\sqrt[3]{8x^3+3}\quad\text{for }x\geq1, \qquad h(x)=e^{4x}\quad\text{for }x\geq k.\)

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

(ii) State the least value of the constant \(k\) such that \(gh(x)\) can be formed.

(iii) Find and simplify an expression for \(gh(x)\).

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

\(f(x)=\operatorname{sec} x,\qquad \frac{\pi}{2}\lt x\lt \frac{3\pi}{2},\)

and

\(g(x)=3(x^2-1),\qquad x\in\mathbb R.\)

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

(a)(ii) Solve \(f^{-1}(x)=\frac{2\pi}{3}\).

(a)(iii) Given that \(gf\) exists, state the domain of \(gf\).

(a)(iv) Solve \(gf(x)=1\).

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

\(h(x)=\ln(4-x),\qquad x\lt 4.\)

Sketch, on the same diagram, the graphs of \(y=h(x)\) and \(y=h^{-1}(x)\), showing clearly any asymptotes and any intersections with the axes.

(a) It is given that \(f:x\mapsto(3x+1)^2-4\), for \(x\geq a\), and that \(f^{-1}\) exists.

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

(ii) Using this value of \(a\), write down the range of \(f\).

(iii) Using this value of \(a\), sketch the graphs of \(y=f(x)\) and \(y=f^{-1}(x)\), stating the intercepts with the coordinate axes.

(b) It is given that \(g(x)=\ln(2x^2+5)\), for \(x\geq0\), and \(h(x)=3x-2\), for \(x\geq0\).

Solve the equation \(hg(x)=4\), giving your answer in exact form.

(a) The function \(f\) is such that \(f(x)=\ln(5x+2)\), for \(x\gt a\), where \(a\) is as small as possible.

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

(ii) Hence find the range of \(f\).

(iii) Find \(f^{-1}(x)\), stating its domain.

(iv) Sketch the graphs of \(y=f(x)\) and \(y=f^{-1}(x)\), stating the exact values of the intercepts of the curves with the coordinate axes.

(b) The function \(g\) is such that \(g:x\mapsto x^{1/2}-4\), for \(x\gt 0\). Solve the equation \(g^2(x)=-2\).

The function \(f\) is defined for \(x\geq0\) by

\(f(x)=5-2\mathrm e^{-x}.\)

(a)(i) Find the domain of \(f^{-1}\).

(a)(ii) Solve

\(f^{-1}(x)=\sqrt{5x-4}.\)

(a)(iii) Sketch, on the same diagram, the graphs of \(y=f(x)\) and \(y=f^{-1}(x)\), showing the intercepts and asymptotes.

(b) The function \(g\) is defined for \(0\leq x\leq0.2\) by

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

Find and simplify an expression for \(f^{-1}g(x)\).

The function \(f\) is defined by

\(f(x)=\ln(2x+1)\qquad\text{for }x\geq0.\)

(a) Sketch the graph of \(y=f(x)\) and hence sketch the graph of \(y=f^{-1}(x)\).

The function \(g\) is defined by

\(g(x)=(x-4)^2+1\qquad\text{for }x\leq4.\)

(b)(i) Find an expression for \(g^{-1}(x)\) and state its domain and range.

(b)(ii) Find and simplify an expression for \(fg(x)\).

(b)(iii) Explain why the function \(gf\) does not exist.

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

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

(a) Explain why \(\mathrm{f}^{-1}\) exists.

(b) On the same axes, sketch the graphs of \(y=\mathrm{f}(x)\) and \(y=\mathrm{f}^{-1}(x)\), giving the coordinates of the vertex of each graph and the intercepts on the coordinate axes.

\(f(x)=3e^{2x}+1\quad\text{for }x\in\mathbb{R}\)

\(g(x)=x+1\quad\text{for }x\in\mathbb{R}\)

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

(ii) Evaluate \(fg^2(0)\).

(iii) On the axes below, sketch the graphs of \(y=f(x)\) and \(y=f^{-1}(x)\), stating the coordinates of the points where the graphs meet the coordinate axes.

A function \(f\) is defined, for \(x\leqslant \dfrac32\), by \(f(x)=2x^2-6x+5\).

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

(ii) On the same axes, sketch the graphs of \(y=f(x)\) and \(y=f^{-1}(x)\), showing the geometrical relationship between them.

(iii) Using your answer from part (i), find an expression for \(f^{-1}(x)\), stating its domain.

The four graphs are labelled \(A\), \(B\), \(C\) and \(D\).

(i) Write down the letter of each graph that represents a function, giving a reason for your choice.

(ii) Write down the letter of each graph that represents a function which has an inverse, giving a reason for your choice.

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

\(f(x)=9\sqrt{x-1}, \qquad g(x)=x^2+2.\)

(i) Find an expression for \(f^{-1}(x)\), stating its domain.

(ii) Find the exact value of \(fg(7)\).

(iii) Solve \(gf(x)=5x^2+83x-95\).