Mathematical functions

What the exam asks

Functions are examined through the other disciplines, not on their own. You will meet them as a graph to classify, a table whose pattern names the relationship, or a formula supplied in the stem that you must substitute into or rearrange. Trigonometry appears almost only in resolving forces, in projectile components, and in Snell's law. The trap that catches most candidates is treating every straight line as a direct proportion: a line with a non-zero intercept is linear but not proportional, so doubling the input does not double the output, and any factor reasoning applied to it is wrong. Before you say 'double x and y doubles', check that the line goes through the origin. If the stem gives an intercept, it gave it to you for a reason.

A function is a rule with one output per input, and Section III wants two things from you: put numbers through it correctly, and recognise the pattern it describes when you meet it as a graph or a table. That is the whole ask. You are not being examined on domains, transformations or identities, and no question will require you to prove anything about a function.

Pattern recognition is the higher-value half. A handful of shapes account for nearly every relationship the exam draws. Linear adds a fixed amount per step. Direct proportion is linear through the origin, which is the only case where doubling the input doubles the output. Inverse proportion halves the output when the input doubles. Inverse square quarters it, and it is everywhere in physics because fields and intensities spread over a sphere. Exponential multiplies by a fixed factor per step. A log is the inverse of an exponential, so it rises without limit but ever more slowly, and it is undefined at zero and below.

The trigonometric ratios are needed for one job in this exam: splitting a quantity into components, or reading an angle in optics. Carry sine, cosine and tangent as ratios of sides, carry the values at 0, 30, 45, 60 and 90 degrees, and carry one technique that outranks all of it. When you cannot recall whether a component takes sine or cosine, test the formula at an angle where you already know the answer. It settles the question in about four seconds and it never fails.

What to hold

• A function gives exactly one output per input, which is why a vertical line can cross its graph at most once.
• A linear relationship adds a constant amount per unit step, and its graph has constant slope equal to rise over run between any two points.
• Direct proportion is the special case of linear that passes through the origin, and only then does doubling the input double the output.
• Inverse proportion means the product of the two variables is constant, so doubling one halves the other.
• An inverse square relationship falls by the square of the factor, so tripling the distance divides the intensity by nine.
• Exponential multiplies by a fixed factor per fixed step, whereas a power law raises the input itself to a power: check whether the output jumps on equal additions or on equal multiplications of the input.
• A log function is the inverse of an exponential: it crosses the horizontal axis at 1, rises without bound but ever more slowly, and is undefined for zero and for negative inputs.
• Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent, which also makes tangent equal to sine over cosine.
• Worth carrying: sine of 30 degrees is one half, sine of 45 is about 0.71, sine of 60 is about 0.87, and cosine runs the same values in reverse.
• The side of a right triangle next to the angle takes cosine and the side across from it takes sine, so on a slope of angle theta the component of weight along the slope is mg sine theta and the component pressing into the surface is mg cosine theta.
• To settle whether a component uses sine or cosine, substitute an angle whose answer you know, such as zero degrees, and keep the formula that gives the right answer there.
• Sine squared plus cosine squared equals 1 for any angle, which is the check that two components you have resolved are consistent.