Review of Functions

The central idea

A function assigns each input in its domain exactly one output. Important data include domain, range, intercepts, zeros, symmetry, transformations, composition, and inverse behavior. The composition $(f\circ g)(x)=f(g(x))$ requires $x$ in the domain of $g$ and $g(x)$ in the domain of $f$.

Definitions, hypotheses, and notation

Domain restrictions survive algebraic simplification because a function includes both a rule and its permitted inputs. The expressions $(x^2-1)/(x-1)$ and $x+1$ agree where the first is defined, but they are not identical functions on the real numbers. This distinction later separates a removable discontinuity from an ordinary point on a line.

Inverse functions interchange inputs and outputs, so their graphs reflect across $y=x$. A horizontal-line test identifies whether an inverse can be a function. Restricting $x^2$ to $x\ge0$ yields the inverse $\sqrt x$; without that restriction, the symbol $\pm\sqrt x$ would assign two outputs and fail the function definition.

Conceptual meaning

A formula is only one representation of a function; graphs, tables, mappings, and verbal rules may describe the same relationship. Calculus statements are always tied to domain: limits approach domain boundaries, derivatives require nearby inputs, and integrals accumulate across intervals.

A dependable method and decision rule

1. Determine the natural domain before simplifying.
2. Identify transformations from a familiar parent function.
3. Find intercepts and use algebra or symmetry to describe shape.
4. For composition, work from the inside outward and enforce both domain conditions.
5. For an inverse, verify one-to-one behavior or restrict the domain first.

Fully worked example

Graphical or geometric meaning

The graph of $f(x)=\sqrt{x-1}$ is the square-root parent shifted right one. Horizontal and vertical transformations affect inputs and outputs differently: $f(x-h)$ shifts right by $h$, while $f(x)+k$ shifts up by $k$.

Common mistakes and why they fail

Verification and reasonableness checks

• Substitute sample inputs from and outside the proposed domain.
• Verify inverses using both compositions.
• Compare algebraic transformations with key graph points.

Domain restrictions survive simplification

A quotient excludes denominator zeros, an even root needs a nonnegative radicand, a real logarithm needs a positive argument, and a composition requires the inner output to lie in the outer domain. Determine restrictions from the original expression. Simplifying $(x^2-1)/(x-1)$ to $x+1$ does not restore $x=1$. For graph transformations, identify the base graph, handle changes inside the input, then vertical scaling and translation outside. Test one distinctive point, intercept, or asymptote afterward. That check often exposes a reversed horizontal shift or an omitted restriction, both of which can be hidden by otherwise correct algebra.

Practice

1. Find the domain of $1/(x-3)$.
2. If $f(x)=2x+1$, find $f^{-1}(x)$.
3. Is $x^2$ one-to-one on all real numbers?

Connections and next steps

• Limits
• Continuity
• Derivative