The diagram shows a region enclosed by three straight lines A, B and C.
The equation of line A is $2y+2=-x$. (a) Show that the equation of the line C is $2y=3x+6$.

(b) Write down three inequalities which satisfy the shaded region.

(c) For a point (x, y) in the shaded region, find the minimum value of x − y.

In the diagram, the coordinates of B and D are (2, 3) and (0, −4½).
OB is parallel to DC. (a) Show that the point C is (3, 0).

(b) Find the equation of line CD.

(c) Given that the length of BC is $\sqrt{r}$, find the value of r.

The diagram shows a straight line AB of gradient $-\frac{3}{4}$. Find:
(a) The coordinates of A,

(b) The distance AB,

(c) The equation of a line parallel to AB passing through (4, 2).

Points (x, 4) and (5, 1) are 5 units apart.
Find x.

$f\left(x\right)=2x-3$ and $g\left(x\right)=4x$.
Find:
(a) $g\left(0.5\right)$

(b) $gf\left(x\right)$

It is given that $\sqrt[3]{5}=1.7$, $\sqrt[3]{50}=3.7$, and $\sqrt[3]{500}=7.9$.
Find the value of $\sqrt[3]{0.005}$.

(a) Find the equation of the line l.

(b) Draw the line on the same grid that is parallel to line l passing through the point (1, 1).

The diagram shows the graph of $f\left(x\right)=-x^2+7x+8$ and $g\left(x\right)=-3x+24$.
f(x) cuts the x-axis at A and B.
f(x) and g(x) intersect at B and D.
F is a point on f(x) and E is a point on g(x).
EF is parallel to the y-axis. (a) Write the coordinates of point C.

(b) Determine the coordinates of A and B.

(c) Find the x-coordinate of point D.

(d) Determine EF in terms of x.

$f\left(x\right)=\frac{3x+2}{2}$
Find:
(i) $\frac{1}{2}f\left(x\right)$,

(ii) f(2).

(a) Complete the table of values for $y=\frac{2}{x};x\ne 0$.

x	-5	-4	-2	-1	-0.5	-0.25	0.25	0.5	1	2	4	5
y	-0.4		-1	-2	-4			4	2	1		0.4

(b) Explain why the value of x cannot be zero.

(c) On the grid, draw the graph of $y=\frac{2}{x}$ for $-5\le x\le 5$.

(d) On the same axes, draw the graph of $y=2x$ for $-5\le x\le 5$.

(e) Use your graphs to solve $\frac{2}{x}-2x=0$.

(f) By drawing a suitable tangent, estimate the gradient of the graph at x = 2.

(a) $f\left(x\right)=\frac{1}{x}$
Find:
(i) f(3),

(ii) $f\left(\frac{k}{4}\right)$, in terms of k,

(iii) k if, $f(2-\frac{k}{4})=2$,

(iv) $ff(8^k-\frac{k}{5})$, in terms of k,

(b) Another function g(x) is such that:
g(192) = 850, and g⁻¹(x) exists.
Find g⁻¹g(192).

(c) Simplify f(x)f(x) − ff(x).
Give your answer as a single fraction, in terms of x.

$f\left(x\right)=3-2x+2$
(a) Find:
(i) f(−1),

(ii) f⁻¹(x).

(b) Find gf(x) in its simplest form.

(c) Solve the equation.
$f(x^2+1)=-17$

The table shows some values of x and y for the function $y=2^{(3+x)}$.

x	-5	-4	-3	-2	-1	0	1
y	0.3		1	2	4	8	16

(a) Complete the table.

(b) On the grid provided plot the points in the table and join them with a smooth curve.

(c) Use your graph to find the value of x when y = 5.

(d) By drawing a tangent, find the gradient of the curve at x = −1.5.

(e) On the same axes draw the graph of y = 3x + 9.

(f) Use your graphs to find the solution to the equations $2^{(3+x)}-3x-3=6$.

$f\left(x\right)=2x^2-1$, $g\left(x\right)=5x+2$, $h\left(x\right)=\frac{x-4}{3}$
(a) Evaluate f(2).

(b) Find the positive value of x for which f(x) = 17.

(c) Find g²(x).

(d) Work out g⁻¹(x).

(e) Given that f(x) = h(x) reduces to $A{x}^2+B{x}^{-7}=0$, find the value of A and B.

The diagram shows the parabola, f(x), and the straight line, g(x).
Points A, B, C and D are the intercepts on the axes.
E is the point of intersection of the two graphs. (a) D is the image of B after B has been translated two units to the right.
Write the coordinates of point D.

(b) Find the equation of the straight line through C and D.
Give your answer in the form y = mx + c.

(c) Find the equation of the parabola in the form $y=a{x}^2+bx+c$.

(d) Work out the coordinates of point E.

(e) Write down the values of x for which f(x) ≥ g(x).

$f\left(x\right)=2^x$ and $g\left(x\right)=2x+3$
(a) Find:
(i) f(3),

(ii) g⁻¹(x),

(iii) gf(x), in its simplest form.

(b) Solve $f\left(x\right)=8^{x-1}$.

(c) Given that gh(x) = 6x − 1, find h(x).

The table shows some values of $y=x^2+x-2$.

x	-3.5	-2	-1.5	-1	0	0.5	1	1.5	2
y	6.75	0	-1.25	-2	-2	p	0	q	4

(a) Calculate the value of p and the value of q.

(b) On the grid, draw the graph of $y=x^2+x-2$ for $-3.5\le x\le 2$.

(c) Write the equation of the line of symmetry of the graph.

(d) (i) On the same grid, draw the graph of x + y = 2.
(ii) Use your graphs to solve the equation $x^2+2x=4$.

The functions f and g are defined as follows:
$f:x\to \frac{(2x-1)}{x^2-2x-3}-\frac{1}{x-3},x>3$, and $g:x\to 2x-3$.
(a) Show that $f\left(x\right)=\frac{1}{x+1}$.

(b) Find:
(i) f⁻¹(x),

(ii) g⁻¹(x),

(iii) fg(x),

(iv) (fg)⁻¹(x).

(c) Solve fg(x) = $\frac{1}{8}$.

(d) Show that $\left(fg{)}^{-1}\right(x)=g^{-1}{f}^{-1}(x)$.

It is given that f and g are functions of x.
$f\left(x\right)=\frac{1}{x+2}+3$ and $g\left(x\right)=3-2x$
(a) Evaluate:
(i) g(−7)

(ii) fg(3)

(b) Find the value of x for which f(x) = 0.

(c) Express in terms of x:
(i) g⁻¹(x),

(ii) g(x)g(x).

(d) Given that p is another function of x such that p(x) = xⁿ.
Find n if p(x) = p⁻¹(x).

(a) Complete the table of values for $y=\frac{1}{3}{x}^3+x^2-2x$.

x	-4	-3	-2	-1	0	1	2	3
y	2.7		5.3	2.7	0	-0.7		12

(b) On the grid, draw the graph of $y=\frac{1}{3}{x}^3+x^2-2x$ for $-4\le x\le 3$.

(c) Use the graph to solve the equation $y=\frac{1}{3}{x}^3+x^2-2x=4$.

(d) The equation $y=\frac{1}{3}{x}^3+x^2-7=5x$ can be solved by drawing a straight line on the grid.
Find the equation of this straight line in the form of y = mx + c.

The table shows some values of $y=x(x+2)(x-1)$ for $-2.5\le x\le 1.5$.

x	-2.5	-2	-1.5	-1	0	1	1.5
y	-4	0		2	0	0

(a) Complete the table.
(b) On the grid, draw the graph of $y=x(x+2)(x-1)$ for $-2.5\le x\le 1.5$.

(c) Use the graph to find the values of x when y = 0.

(d) By drawing a suitable straight line, find one of the solutions for $y=x^3+x^2-x-1$.

(a) f(x) = 6x − x²

(i) Complete the table for f(x).

x	0	1	2	3	4	5	6
f(x)	0			9			0

(ii) On the grid, draw the graph of y = f(x) for 0 ≤ x ≤ 6.

(iii) Use your graph to solve the equation f(x) = 6.

(iv) By drawing a tangent at the point where x = 4, estimate the gradient of the graph of y = f(x) when x = 4.

(b) g(x) = 2^x

(i) Complete the table for g(x).

x	0	1	2	3	3.3
g(x)				8	9.8

(ii) On the same grid opposite, draw the graph of y = g(x) for 0 ≤ x ≤ 3.3.

(c) Use your graphs to find x when f(x) = g(x).

The table shows values of the function $y=\frac{1}{3}{x}^3-x+4$.

X	-3	-2.5	-2	-1.5	-1	0	1.5	2	2.5	3
y	-2		3.3	4.0	4.7				6.7	10

(a) Complete the table.
(b) On the grid, draw the graph of $y=\frac{1}{3}{x}^3-x+4$ for $-3\le x\le 3$.