This mini-lesson covers SL Functions: notation, domain & range, composite and inverse functions, transformations of graphs, and the key families — quadratic, rational, exponential and logarithmic.
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A function f maps each input x to exactly one output f(x). The domain is the set of allowed inputs; the range is the set of outputs produced.
Vertical line test: a graph is a function only if every vertical line meets it at most once.
Applying one function then another gives a composite. (f ∘ g)(x) = f(g(x)) — do g first.
f(x) = 2x + 1, g(x) = x². Then (f ∘ g)(3) = f(g(3)) = f(9) = 2(9) + 1 = 19.
Order matters: in general f ∘ g ≠ g ∘ f.
The inverse f⁻¹ undoes f: f⁻¹(f(x)) = x. It exists when f is one-to-one. To find it, swap x and y and make y the subject.
f(x) = 3x − 5 ⇒ x = 3y − 5 ⇒ y = (x + 5)/3, so f⁻¹(x) = (x + 5)/3. Check f⁻¹(7) = 12/3 = 4.
Graphically: f⁻¹ is the reflection of f in the line y = x.
Starting from y = f(x):
Inside vs outside: changes inside f( ) act on x (horizontal, and "backwards"); changes outside act on y (vertical).
Tap a mapping, then the transformation it describes.
A quadratic f(x) = ax² + bx + c has a parabola graph. Useful forms and facts:
The vertex (turning point) is at x = −b/2a. Completing the square gives y = a(x − h)² + k with vertex (h, k).
A rational function is a ratio of polynomials. The reciprocal f(x) = 1/x has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0.
Tap a statement on the left, then its matching partner on the right.
f(x) = aˣ (a > 0) models growth/decay; it has a horizontal asymptote y = 0 and passes through (0, 1). Its inverse is g(x) = logₐx.
Always state the domain — it can restrict the range and decide whether an inverse exists. Sketch key features: intercepts, asymptotes, vertices.
Exam habit: for composites, substitute the inner function whole and simplify carefully; for inverses, verify with one value.
Notation: f(x) is one output per input; state the domain
Composite: (f∘g)(x) = f(g(x)) — inner function first
Inverse: swap x and y; reflection in y = x; needs one-to-one
Transformations: +a up, −b right, p·f stretch, −f reflect
Quadratics: Δ = b²−4ac decides roots; vertex at x = −b/2a
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