Course

O Level Additional Mathematics: Functions and Quadratics (4037)

Name: O Level Additional Mathematics: Functions and Quadratics (4037)
Price: 999 BDT

Functions, composite and inverse functions, modulus graphs, and quadratic functions from solving to the discriminant, for Cambridge O Level Additional Mathematics 4037.

31 lessons5h 38mEnglish

What you'll learn

Use the words function, domain, range, one-one, many-one, inverse and composite correctly
Use function notation: f(x), f : x ↦ ..., f⁻¹(x), fg(x) and f²(x)
Find the domain and range of a function, a composite function and an inverse function
Form composite functions and explain why fg may not equal gf
Sketch y = |f(x)| when f(x) is linear or quadratic
Explain why a function has no inverse, find the inverse of a one-one function and sketch both graphs
Solve quadratic equations by factorisation, the formula and completing the square
Find the maximum or minimum value of a quadratic and use it to sketch the graph and find a range
Solve quadratic inequalities and write the solution set correctly
Use the discriminant to decide the roots of a quadratic and whether a line cuts, touches or misses a curve

Requirements

Cambridge O Level Mathematics 4024 or equivalent: expanding brackets, factorising, and solving linear equations

Course content

31 lessons · 5h 38m

1.1-1.8 · Functions

Mappings and functions, domain and range, one-one and many-one functions, composite functions, modulus graphs, inverse functions and their graphs.

0/14

Mappings

A mapping joins each input to its outputs. It can be one-one, one-many, many-one or many-many.

Preview10m 2.

Definition of a function

A function gives each input exactly one output. Write it as f(x) = ... or f : x ↦ ...

Preview10m

Domain and range

The domain is the set of inputs you may use. The range is the set of outputs the function really gives.

12m

One-one and many-one functions

A one-one function never gives the same output twice. A horizontal line shows it on a graph.

10m

Composite functions

In fg(x), g acts first and f acts second. fg is usually not the same as gf, and f²(x) means f(f(x)).

12m

Domain and range of composite functions

fg exists when every output of g is an input f accepts. Then find its range and solve fg(x) = k.

14m

Modulus functions

The modulus |x| is the distance of x from 0, so it is never negative. |f(x)| keeps f(x) when it is positive.

10m

Graphs of y = |f(x)| where f(x) is linear

Draw y = f(x), then reflect the part below the x-axis upward. The result is a V shape.

12m

Functions that have no inverse

A many-one function cannot be undone, because one output came from two inputs. Say this in words.

10m

10.

Inverse functions

Write y = f(x), swap x and y, and make y the subject. The domain of f⁻¹ is the range of f.

12m

11.

The graph of a function and its inverse

The graph of f⁻¹ is the reflection of the graph of f in the line y = x.

10m

12.

Functions: practice

A timed drill of mixed questions on every idea in this chapter, from easy to exam level.

15m

13.

Functions: exam practice

Past-paper style questions with full worked solutions and mark schemes.

15m

14.

Functions: summary

Key points, notation and common mistakes for functions on one screen.

2.1-2.5 · Quadratic functions

Solving quadratic equations, completing the square, maximum and minimum values, sketching and ranges, modulus graphs, quadratic inequalities, the discriminant, and lines meeting curves.

0/15

Solving quadratic equations by factorisation

Make one side zero, factorise, and set each bracket equal to zero.

Preview10m 2.

Solving quadratic equations using the formula

The formula solves every quadratic, even one that does not factorise. Give exact surds or 3 s.f.

Preview12m

Completing the square

Write ax² + bx + c as a(x + p)² + q. This form also solves the equation.

12m

Maximum and minimum values of a quadratic function

From a(x + p)² + q, the vertex is (-p, q). If a > 0, q is the minimum; if a < 0, q is the maximum.

12m

Sketching the graph of a quadratic function

Mark the shape, the vertex, the y-intercept and any x-intercepts, then join them with a smooth curve.

12m

The range of a quadratic function for a given domain

Check whether the vertex lies inside the domain, then compare the end values. Write the range with correct notation.

12m

Graphs of y = |f(x)| where f(x) is quadratic

Sketch the parabola, then reflect the part below the x-axis upward. A minimum below the axis becomes a maximum.

12m

Quadratic inequalities

Find the roots, sketch the curve, and read where it is above or below the x-axis.

12m

Roots of quadratic equations

The discriminant b² - 4ac tells you if there are two real roots, two equal roots or no real roots.

10m

10.

Using the discriminant to find an unknown constant

Put the condition on b² - 4ac, then solve the equation or inequality in k.

12m

11.

Intersection of a line and a curve

Substitute the line into the curve to get one quadratic. Its discriminant says if the line cuts or misses.

12m

12.

A line that is a tangent to a curve

A tangent touches the curve at one point, so b² - 4ac = 0. Use it to find k and the point of contact.

12m

13.

Quadratic functions: practice

A timed drill of mixed questions on every idea in this chapter, from easy to exam level.

15m

14.

Quadratic functions: exam practice

Past-paper style questions with full worked solutions and mark schemes.

15m

15.

Quadratic functions: summary

Key points, formulas and common mistakes for quadratic functions on one screen.

About the creator

8bitlesson

Cambridge O Level, A Level and university courses, taught through interactive labs.

Mentora builds the sciences and mathematics as things you can move rather than videos you watch. Cambridge O Level and A Level mathematics, physics, chemistry and accounting, a primary mathematics track, and first-year university computer science. Every lesson works a real exam question end to end.