Functions are mathematical relationships that pop up everywhere in maths,...
Maths182 views·Updated 11 Jul 2026·2 pagesUnderstanding Functions and Graphs for A-Level
maria@maria_reji
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Modulus and Functions Basics
The modulus (or absolute value) always gives you a positive result - think of it as the distance from zero. So |−2| = 2 and |5| = 5. When you see y = |x|, you're looking at a V-shaped graph that never dips below the x-axis.
To solve modulus equations like |3x − 5| = 2 − ½x, you need to consider two cases. First, when the expression inside is positive: 3x − 5 = 2 − ½x, giving x = 2. Second, when it's negative: −(3x − 5) = 2 − ½x, giving x = 6/5.
Functions can be one-to-one (each x-value maps to exactly one y-value, and vice versa) or many-to-one (multiple x-values can give the same y-value). Only one-to-one functions have inverse functions that work properly.
Quick Check: Use the horizontal line test - if any horizontal line crosses the graph more than once, it's many-to-one!
2
of 2
Inverse Functions and Transformations
Inverse functions essentially "undo" what the original function does. The range of f becomes the domain of f⁻¹, and vice versa. To find an inverse, swap x and y, then solve for y.
For example, if f = x² − 3 (where x ≥ 0), let y = x² − 3, rearrange to get x = √, so f⁻¹ = √. The graphs of f and f⁻¹ are reflections of each other across the line y = x.
Function transformations follow predictable patterns: f shifts left by a units, f + a shifts up by a units, f(−x) reflects in the y-axis, and −f reflects in the x-axis. Stretches work differently - f(ax) compresses horizontally by factor 1/a, whilst af stretches vertically by factor a.
Memory Tip: When combining functions like fg, always work from right to left - apply g first, then f!
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Maths182 views·Updated 11 Jul 2026·2 pagesUnderstanding Functions and Graphs for A-Level
maria@maria_reji
Functions are mathematical relationships that pop up everywhere in maths, from solving equations to graphing curves. Understanding modulus functions, inverse functions, and transformations will give you the tools to tackle complex problems with confidence.
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Modulus and Functions Basics
The modulus (or absolute value) always gives you a positive result - think of it as the distance from zero. So |−2| = 2 and |5| = 5. When you see y = |x|, you're looking at a V-shaped graph that never dips below the x-axis.
To solve modulus equations like |3x − 5| = 2 − ½x, you need to consider two cases. First, when the expression inside is positive: 3x − 5 = 2 − ½x, giving x = 2. Second, when it's negative: −(3x − 5) = 2 − ½x, giving x = 6/5.
Functions can be one-to-one (each x-value maps to exactly one y-value, and vice versa) or many-to-one (multiple x-values can give the same y-value). Only one-to-one functions have inverse functions that work properly.
Quick Check: Use the horizontal line test - if any horizontal line crosses the graph more than once, it's many-to-one!
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Inverse Functions and Transformations
Inverse functions essentially "undo" what the original function does. The range of f becomes the domain of f⁻¹, and vice versa. To find an inverse, swap x and y, then solve for y.
For example, if f = x² − 3 (where x ≥ 0), let y = x² − 3, rearrange to get x = √, so f⁻¹ = √. The graphs of f and f⁻¹ are reflections of each other across the line y = x.
Function transformations follow predictable patterns: f shifts left by a units, f + a shifts up by a units, f(−x) reflects in the y-axis, and −f reflects in the x-axis. Stretches work differently - f(ax) compresses horizontally by factor 1/a, whilst af stretches vertically by factor a.
Memory Tip: When combining functions like fg, always work from right to left - apply g first, then f!
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