Domains, composition and radian measure

Try

A circular wheel has radius 4 cm. A marked point moves along an arc of length 6 cm. Before calculating its angle, decide which quantity is a length and which is a ratio of lengths. Now consider a separate machine: it subtracts 2 from an input, then takes the reciprocal. Which input makes the second operation impossible?

Write both answers before continuing. These are two common calculus preparation tasks: track units and check whether an expression is defined. Neither requires a derivative yet.

Predict

Let f(u)=1/u and g(x)=x−2. Predict whether f(g(x)) and g(f(x)) describe the same machine. Test your prediction at x=4. Also predict whether simplifying a fraction can restore an input that originally caused division by zero.

Experiment

Complete the missing entries on paper. “Undefined” is a mathematical result, not a request to put zero in the table.

Input x	Intermediate g(x)=x−2	Output f(g(x))
1	−1	−1
2	0	undefined
3	1	1
4	2	1/2

At x=4, the reverse composition is g(f(4))=1/4−2=−7/4. The order changes both the result and the excluded inputs. Draw two boxes with an arrow from the first output to the second input; label the intermediate value. This diagram is often more useful than immediately expanding an expression.

Understand

The domain is the set of permitted inputs. The real square root of x−1 requires x at least 1. A denominator x−3 excludes 3. Therefore √(x−1)/(x−3) has domain [1,3)∪(3,∞): include 1, exclude 3, and include larger values. The symbols [ and ] include an endpoint; parentheses exclude it. Infinity is never an included real endpoint.

For a composition f(g(x)), first require that g(x) exists, then require that its output belongs to the domain of f. For example, if g(x)=√x and f(u)=1/(u−2), require x at least zero and exclude x=4. Checking only the outer denominator without checking the square root misses negative inputs.

A radian is defined by θ=s/r, where θ is angle in radians, s is arc length and r is positive radius, with matching length units. The wheel’s angle is 6/4=3/2 radians. One full circle has arc length 2πr, so one revolution is 2π radians or 360 degrees. Consequently 60 degrees is π/3 radians. The equation s=rθ assumes radians. Degree numbers cannot be inserted unchanged.

Cancellation is another domain trap: (x²−1)/(x−1)=x+1 only where x≠1. The simpler expression has a value at 1; the original does not. Limits will let us discuss its nearby behavior without pretending that the original domain changed.

Apply

Use the practice bank to evaluate compositions, find exclusions and convert angle units. For written domains, state each restriction before combining them. When someone reports an angle “1.5,” ask which unit is intended. When someone reports a simplified formula, ask whether its original exclusions were retained.

Continue

Next, distinguish the value at an input from nearby behavior. If factoring, fractions or coordinate reading remain unfamiliar, pause for targeted algebra practice; the lessons remain open. This readiness lesson introduces the checks used throughout the sequence, rather than certifying all precalculus skills.