Curriculum · Course · Week 4 · فارسی
Composition connects the output of one function to the input of another. Its domain must satisfy both stages. Inverse trigonometric functions recover a principal angle from a ratio. A graph or calculator cannot choose every angle with the same sine. A restricted range makes the result unique. This week closes the function review before limits begin.
Evaluate the inner function first. Carry its output into the outer rule and preserve all domain restrictions. Use radians in calculus. For real inputs, arcsin accepts [−1,1] and returns [−π/2,π/2]. Arccos returns [0,π]. Arctan returns (−π/2,π/2). Check dimensions before treating a mathematical expression as an engineering model.
Let f(u)=√u and g(x)=x−2. Find f(g(x)).
f(g(x))=√(x−2) requires x≥2. Reversing the order gives √x−2 with x≥0.
Evaluate arcsin(1/2) and arcsin(sin(5π/6)).
Both equal π/6. The principal output is restricted, so the second expression does not return 5π/6.
A calibration has v=0.5x+1. A display reports d=10v. Find the complete model.
d(x)=5x+10. At x=2, the voltage is 2 and the display is 20. The two stages can be checked separately.
The blue curve begins at x=2. The orange curve begins at x=0. Parentheses change which operation is performed first, so the curves have different starting points and domains.
The blue output rises to π/2 and then falls. At 5π/6 it returns π/6. The identity line keeps rising. This shows exactly where restricting the inverse range matters.
For each problem, write the domain or units before calculating. After checking the answer, change one numerical value and solve the variant without looking back.
For h=f∘g, the domain contains exactly the inputs x for which g(x) is defined and its output belongs to the domain of f. Checking only the outer formula can miss a restriction from the inner stage. Composition is generally not commutative. It is associative when the expressions are defined, so a three-stage calibration can be grouped without changing the order of operations.
Let g(x)=√(x−1) and f(u)=u². Algebra gives f(g(x))=x−1, but the original composition still requires x≥1. By contrast, g(f(x))=√(x²−1) requires x≤−1 or x≥1. The two answers differ in both rule and domain. Even when a simplified expression looks harmless, keep the conditions established before simplification.
Sine repeats and is not one-to-one on the real line. Restricting sine to [−π/2,π/2] makes it invertible. Arcsine therefore accepts [−1,1] and returns only that principal angle interval. Similarly, arccosine returns [0,π] and arctangent returns (−π/2,π/2). The identity sin(arcsin u)=u holds for −1≤u≤1, but arcsin(sin θ)=θ holds only when θ is in arcsine’s principal interval.
Solve sin θ=1/2 on 0≤θ<2π. The principal result is π/6, but sine is also positive in the second quadrant. The second solution is π−π/6=5π/6. There are exactly two solutions on this interval. A calculator’s arcsine button gives the principal answer, not a complete solution set. For cos θ=−√2/2 on the same interval, the solutions are 3π/4 and 5π/4.
A sensor follows v(T)=0.02T+0.5 V on 0≤T≤100 °C. A display applies d(v)=100v counts, but cannot show more than 200 counts. The unsaturated composition is d(T)=2T+50. The cap is reached when 2T+50=200, so T=75 °C. The actual model is d(T)=2T+50 for 0≤T≤75 and d(T)=200 for 75<T≤100. Above 75 °C, different temperatures give the same displayed value, so the display cannot be uniquely inverted there. This is information loss caused by saturation, not a failure of algebra.
First check domain, units and endpoint values. Then check a few interior points and compare predictions with the stated operating conditions. In a formula such as e−t/τ, t and τ must use matching time units. In ln(v/Vref), the ratio is dimensionless. Neither dimensional consistency nor an attractive graph proves that a model matches the physical system; agreement with appropriate observations is also needed.
Problems 1–6 complete the core function-review checkpoint; 7–8 are optional.
Common mistakes: losing inner-domain restrictions, reporting only a principal calculator angle when all solutions are requested, and trying to invert a saturated model uniquely. You are ready for limits when you can explain the domain and behaviour of each stage without relying on a plot alone.
An original summary aligned with these textbook topics. References are optional further reading.