Topic 1 of 9 · Form 3
Functions
Domain, range, one-one and many-one mappings; composite functions fg(x) = f(g(x)); inverse functions f⁻¹ (only for one-one functions); the graphs of f and f⁻¹ are reflections in y = x.
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Definitions
Domain, range, one-one and many-one mappings; composite functions fg(x) = f(g(x)); inverse functions f⁻¹ (only for one-one functions); the graphs of f and f⁻¹ are reflections in y = x.
Worked example: f(x) = 2x + 3 and g(x) = x² − 1. Find fg(x), gf(2) and f⁻¹(x).
fg(x) = 2(x² − 1) + 3 = 2x² + 1
gf(2) = g(7) = 48
y = 2x + 3 → f⁻¹(x) = (x − 3)/2
Modulus functions
|f(x)| reflects parts of the graph below the x-axis. Solve |ax + b| = c by considering ax + b = c and ax + b = −c; solve |ax + b| = |cx + d| by squaring both sides.
Worked example: Solve |2x − 1| = 5.
2x − 1 = 5 → x = 3; 2x − 1 = −5 → x = −2
Graphs
Sketch y = |f(x)| for linear and quadratic f; know the shapes of cubic, exponential and logarithmic graphs.
Check yourself
- f(x) = x² + 2, x ≥ 0. Find the range of f and f⁻¹(x).
- Solve |x + 3| = |2x|.
- Why does f(x) = x², x ∈ ℝ, not have an inverse?
Show answers
- Range f(x) ≥ 2; f⁻¹(x) = √(x − 2), x ≥ 2
- (x + 3)² = 4x² → 3x² − 6x − 9 = 0 → x = 3 or x = −1
- It is many-one (e.g. f(2) = f(−2)).
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