Precalculus: Functions & Transformations
Precalculus builds the bridge to calculus. Master function notation, transformations, exponential and logarithmic functions, and composite/inverse functions.
Key Concepts
Function Notation
f(x) means 'the value of f at x'. f(3) = plug in x=3. Domain = allowed inputs. Range = possible outputs. A function passes the vertical line test — each x gives exactly one y.
Transformations
f(x)+k shifts up k. f(x−h) shifts right h. af(x) stretches vertically by a. f(bx) compresses horizontally by factor b. −f(x) reflects over x-axis.
Exponential & Logarithms
Exponential: f(x) = aᵇˣ. Models growth/decay. log_b(x) is the inverse: log_b(bˣ) = x. Natural log ln(x) uses base e ≈ 2.718. ln(eˣ) = x.
Inverse & Composite Functions
Inverse f⁻¹ undoes f: if f(3)=7 then f⁻¹(7)=3. Composite (f∘g)(x) = f(g(x)): apply g first, then f. (f∘f⁻¹)(x) = x always.
Live Python Practice
Interactive Lab
Check Your Understanding
f(g(x)) means:
The graph of f(x)+3 compared to f(x) is:
log_b(b^x) equals: