Functions, Logarithms, Trigonometry and Probability
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Part 1 of 1
1. Functions and their domains
Most function questions on this paper are really domain questions in disguise. Three things restrict a domain and they are the same three every time.
You cannot divide by zero, so anything in a denominator must be excluded where it vanishes. You cannot take an even root of a negative number, so expressions under a square root must be greater than or equal to zero. You cannot take the logarithm of zero or a negative number, so log arguments must be strictly positive.
Beyond that, know the standard shapes. A linear function y = mx + q is a straight line of gradient m. A quadratic is a parabola opening upwards when the coefficient of x squared is positive, with its vertex at x = negative b over 2a. Recognising the shape usually answers the question before any algebra begins.
Worked example
Find the domain of f(x) = square root of (x minus 2), divided by (x minus 5). The root needs x minus 2 greater than or equal to 0, so x is at least 2. The denominator needs x not equal to 5. The domain is therefore x greater than or equal to 2, excluding 5.
1.The argument of a logarithm must be strictly greater than zero.
2.A parabola with a negative coefficient of x squared opens upwards.
3.A function can return only one output value for each input.
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