A CONCISE INTRODUCTION TO PURE
MATHEMATICS LIEBECK 4TH EDITION
SOLUTION MANUAL 2026 STUDY GUIDE
WITH ANSWERS CHAPTERS 1 TO 26
⩥ Domain. Answer: The set of possible inputs for a function.
⩥ Range. Answer: The set of possible outputs of a function.
⩥ Discriminant. Answer: b² - 4ac > 0 then two distinct real roots.
b² - 4ac = 0 then one repeated real root.
b² - 4ac < 0 then a quadratic function has no real roots.
⩥ Types of Lines for Regions. Answer: If y < f(x) or y > f(x) then the
curve y = f(x) is not included in the region, and is represented by a
dotted line.
If y ≤ f(x) or y ≥ f(x) then the curve y = f(x) is included in the region,
and is represented by a solid line.
,⩥ Graph Translations. Answer: y = f(x) + a is a translation of the graph y
= f(x) by a upwards.
y = f(x + a) is a translation of the graph y = f(x) by a to the left.
⩥ Graph Stretches. Answer: y = af(x) is a stretch of the graph y = f(x) by
a scale factor of a in the vertical direction.
y = f(ax) is a stretch of the graph y = f(x) by a scale factor of 1/a in the
horizontal direction.
⩥ Graph Reflections. Answer: y = -f(x) is a reflection of the graph of y =
f(x) in the x-axis.
y = f(-x) is a reflection of the graph of y = f(x) in the y-axis.
⩥ Gradient of Equation. Answer: m = (y₂ - y₁) ÷ (x₂ - x₁)
⩥ Equation of a Line. Answer: y - y₁ = m(x - x₁)
with coords (x₁, y₁)
⩥ Distance Formula. Answer: √((x₂ - x₁)² + (y₂ - y₁)²)
, from (x₁, y₁) to (x₂, y₂)
⩥ Perpendicular Bisector. Answer: -1/m
where m is original gradient
⩥ Standard Equation of a Circle. Answer: (x - a)² + (y - b)² = r²
with centre (a, b) and radius r
⩥ Equation of a Circle (fg). Answer: x² + y² + 2fx + 2gy + c = 0
with centre (-f, -g) and radius √(f² + g² - c)
⩥ Circle Theorems. Answer: • Tangent to a circle is perpendicular to the
radius of the circle at the point of intersection.
• Perpendicular bisector of a chord will go through the circle centre.
• If triangle forms across the circle, its diameter is the hypotenuse of the
right-angled triangle.
• Equations of the perpendicular bisectors of two different chords will
intersect at the circle centre.
MATHEMATICS LIEBECK 4TH EDITION
SOLUTION MANUAL 2026 STUDY GUIDE
WITH ANSWERS CHAPTERS 1 TO 26
⩥ Domain. Answer: The set of possible inputs for a function.
⩥ Range. Answer: The set of possible outputs of a function.
⩥ Discriminant. Answer: b² - 4ac > 0 then two distinct real roots.
b² - 4ac = 0 then one repeated real root.
b² - 4ac < 0 then a quadratic function has no real roots.
⩥ Types of Lines for Regions. Answer: If y < f(x) or y > f(x) then the
curve y = f(x) is not included in the region, and is represented by a
dotted line.
If y ≤ f(x) or y ≥ f(x) then the curve y = f(x) is included in the region,
and is represented by a solid line.
,⩥ Graph Translations. Answer: y = f(x) + a is a translation of the graph y
= f(x) by a upwards.
y = f(x + a) is a translation of the graph y = f(x) by a to the left.
⩥ Graph Stretches. Answer: y = af(x) is a stretch of the graph y = f(x) by
a scale factor of a in the vertical direction.
y = f(ax) is a stretch of the graph y = f(x) by a scale factor of 1/a in the
horizontal direction.
⩥ Graph Reflections. Answer: y = -f(x) is a reflection of the graph of y =
f(x) in the x-axis.
y = f(-x) is a reflection of the graph of y = f(x) in the y-axis.
⩥ Gradient of Equation. Answer: m = (y₂ - y₁) ÷ (x₂ - x₁)
⩥ Equation of a Line. Answer: y - y₁ = m(x - x₁)
with coords (x₁, y₁)
⩥ Distance Formula. Answer: √((x₂ - x₁)² + (y₂ - y₁)²)
, from (x₁, y₁) to (x₂, y₂)
⩥ Perpendicular Bisector. Answer: -1/m
where m is original gradient
⩥ Standard Equation of a Circle. Answer: (x - a)² + (y - b)² = r²
with centre (a, b) and radius r
⩥ Equation of a Circle (fg). Answer: x² + y² + 2fx + 2gy + c = 0
with centre (-f, -g) and radius √(f² + g² - c)
⩥ Circle Theorems. Answer: • Tangent to a circle is perpendicular to the
radius of the circle at the point of intersection.
• Perpendicular bisector of a chord will go through the circle centre.
• If triangle forms across the circle, its diameter is the hypotenuse of the
right-angled triangle.
• Equations of the perpendicular bisectors of two different chords will
intersect at the circle centre.
