1.2 Functions

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.

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 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.

If a graph folds back, restrict the original domain to a one-one branch before defining an inverse. The chosen branch determines which inverse values are allowed.

For x², choosing x≥0 gives inverse √x; choosing x≤0 gives inverse −√x. State the restriction as part of the function, not as an afterthought.

On [0,∞), f(x)=x² has f⁻¹(x)=√x with domain x≥0. On (−∞,0], the inverse is −√x.

Writing ±√x gives two outputs and is an inverse relation, not a single-valued inverse function.

For y=f(x), y=f(x)+a shifts up a, y=f(x−a) shifts right a, y=bf(x) scales vertical values, and y=f(bx) scales horizontal coordinates by 1/b.

Read transformations from the formula, track a known point (x,y), and apply inside changes before outside changes when several are combined.

From y=x², y=2(x−3)²+1 has vertex (3,1), vertical stretch 2 and the same upward-opening shape.

The sign inside the bracket reverses the horizontal direction: f(x−3) moves right, not left.