1.2 Functions
- Syllabus
- 9709–2028–2029
- Topic
- 1.2
- Level
- AS
Learning objectives
1.2.1Function terminology• understand the terms function, domain, range, one-one function, inverse function and composition of functions1.2.2Range and composition• identify the range of a given function in simple cases, and find the composition of two given functions e.g. range of: x x 1f 7 for x 1H and range of: xx 1g 27 + for x R!. Including the condition that a composite function gf can only be formed when the range of f is within the domain of g.1.2.3One-one and inverse functions• determine whether or not a given function is one-one, and find the inverse of a one-one function in simple cases e.g. finding the inverse of: x x23 4h 27 -+^h for <x 2 3 -1.2.4One-one and inverse functions• illustrate in graphical terms the relation between a one-one function and its inverse Sketches should include an indication of the mirror line y = x.1.2.5Graph transformations• understand and use the transformations of the graph of y = f(x) given by y = f(x) + a, y = f(x + a), y = af(x), y = f(ax) and simple combinations of these. Including use of the terms 'translation', 'reflection' and 'stretch' in describing transformations. Questions may involve algebraic or trigonometric functions, or other graphs with given features.
Function notation records input, output and the allowed domain
A function f maps each permitted input x to exactly one output f(x). The domain is the allowed set of inputs; the range is the set of outputs actually produced.
Read f(a) as the output when input a is used. For composites, (f∘g)(x)=f(g(x)) and x must lie in g’s domain with g(x) in f’s domain.
For f(x)=√(x−2), the real domain is x≥2 and the range is y≥0; f(6)=2.
A relation can assign one input several outputs and then is not a function; domain restrictions are part of the definition, not optional notes.
Range and composition depend on the actual domain of each function
The range is the set of outputs produced by a function on its stated domain. For (f∘g)(x)=f(g(x)), x must be allowed by g and g(x) must lie in f’s domain.
Find the inner range first, then apply the outer function. Restricting a domain can change the range and can make a formula that is usually invertible behave differently.
If g(x)=x² on [0,2] and f(u)=√u, then (f∘g)(x)=x on [0,2]; using all real x would describe a different domain and range.
The range of f∘g is not automatically the range of f; only the part of f reached by g is relevant.
A one-one function passes the horizontal-line test and has an inverse on its range
A function is one-one if f(a)=f(b) implies a=b. Equivalently, no horizontal line meets its graph more than once. This allows an inverse function to undo f on its range.
To find f⁻¹, write y=f(x), interchange x and y, then solve for y. The inverse domain is the original range, and its range is the original domain.
f(x)=3x−2 is one-one on ℝ and f⁻¹(x)=(x+2)/3. The graphs reflect in y=x.
A function can have an inverse relation but not an inverse function if it is not one-one on the stated domain.
An inverse graph swaps every coordinate and reflects in $y=x$
If (a,b) lies on y=f(x), then (b,a) lies on y=f−1(x). Swapping coordinates reflects the entire graph in the mirror line y=x.
Draw y=x, reflect several defining points and preserve their order. An x-intercept (a,0) becomes the y-intercept (0,a); horizontal and vertical asymptotes swap; the domain and range swap.
The two graphs can meet only on y=x, because a reflected point is unchanged there. Such intersections satisfy f(x)=x, provided x lies in both relevant domains.
Use the original graph's endpoints, open/closed points, turning behaviour and asymptotes rather than inventing a new shape. The inverse must still pass the vertical-line test because the original is one-one.
Do not reflect in the x-axis or y-axis. A correct inverse sketch must show the relevant mirror line y=x and swap coordinates, not merely reverse the curve's direction.
Outside changes act on $y$; inside changes act inversely on $x$
| New graph | Coordinate mapping from (x,y) | Description |
|---|---|---|
| y=f(x)+a | (x,y+a) | translate by (0,a) |
| y=f(x+a) | (x−a,y) | translate by (−a,0) |
| y=af(x) | (x,ay) | stretch parallel to y-axis, factor ∣a∣; reflect in x-axis if a<0 |
| y=f(ax) | (x/a,y) | stretch parallel to x-axis, factor 1/∣a∣; reflect in y-axis if a<0 |
Track a known point, intercept, asymptote or turning point through the coordinate mapping. This is safer than relying on a memorised verbal direction, especially for inside transformations.
For $y=3f(-2x)+4$: first map $x\mapsto -x/2$ (horizontal reflection and factor $1/2$), then $y\mapsto3y+4$ (vertical factor $3$, then translate up $4$).
When two transformations share one axis, order can change the translation size. State a valid sequence explicitly and verify it by applying the operations to the function or to a reference point.
Inside signs and factors act inversely: f(x−3) moves right 3, while f(3x) has horizontal scale factor 1/3. Use the syllabus terms translation, reflection and stretch with axis/direction and factor.