Summary — What This Chapter Covers
- A function maps each input in its domain to exactly one output; the set of outputs is the range.
- gf(x) means "do f first, then g". It exists only when the range of f lies inside the domain of g.
- Only one-one functions have inverses. The domain of f−1 is the range of f, and the range of f−1 is the domain of f.
- The graph of y = f−1(x) is the reflection of y = f(x) in the line y = x.
- f(x) + a, f(x + a), af(x) and f(ax) are translations and stretches; describe them precisely.
1. Domain, Range and Composite Functions
Composition
What goes in, what comes out
The domain is the set of inputs you are allowed to use; the range is the set of outputs that actually occur. Find the range by thinking about the graph over the given domain, and use a completed square for quadratics.
A function is one-one if each output comes from exactly one input (a horizontal line meets the graph at most once). The composite gf(x) = g(f(x)) applies f first.
Composite rule
gf exists only if range of f ⊆ domain of g. gf(x) is usually different from fg(x).
Worked Example
f(x) = 2x − 1 and g(x) = x2 + 3, both for x ∈ ℝ. Find fg(x) and gf(x), and state the range of gf.
Practice Question
f(x) = 1/x for x ≥ 1. State the range of f.
2. Inverse Functions
Inverses
Swap the roles of x and y
To find f−1, write y = f(x), rearrange to make x the subject, then swap x and y. The domain of f−1 is the range of f. If the function is not one-one on ℝ (a quadratic, for example), it must be restricted first; the restriction tells you which square root to take.
On a sketch, y = f(x) and y = f−1(x) are reflections of each other in y = x; always draw and label that mirror line.
Worked Example
f(x) = (x − 2)2 + 3 for x ≥ 2. Find f−1(x) and state its domain.
Practice Question
f(x) = (3x + 1)/(x − 2) for x ≠ 2. Find f−1(x).
Sketching f and f−1
Points where y = f(x) meets y = f
−1(x) lie on y = x, so solving f(x) = x is often the quickest way to find them.
3. Transformations of Graphs
Transformations
Translations, reflections and stretches
Changes outside the function act on y and behave as you expect. Changes inside the bracket act on x and do the opposite of what they look like.
The four basic transformations
Worked Example
Describe a sequence of transformations mapping y = f(x) onto y = 2f(x − 1) + 3. The point (4, 5) lies on y = f(x); find its image.
Memorise