Definitions are choices with a price

Mathematics · Mathematical Thinking · ages 22-24

Name ______________________ Date ____________

f(x) equals x squared over x and g(x) equals x simplify to the same expression, so they are the same function.
Circle one: True False
f(x) = x squared over x and g(x) = x. On the integers from -2 to 2, at how many points do f and g agree?
Answer: ______________
f(x) = x squared over x and g(x) = x simplify to the same expression. Are they the same function?
Circle one: True False
One definition says a function is a rule; another says it is a set of input-output pairs. What does the pairs version make especially easy?
- Deciding whether two formulas define the same function
- Evaluating formulas faster
- Drawing smoother graphs
Let f(x) = 1 if x is rational and 0 otherwise. What is f(0) + f(sqrt(2)) + f(1/2)?
Answer: ______________
A subspace must be nonempty. Sam says the empty set already satisfies closure under addition and scaling vacuously, so nonempty exists only to rule it out. Is Sam right?
Circle one: True False
Thomae's function is 0 at irrationals and 1/q at p/q in lowest terms. Is it continuous at x = sqrt(2)/2?
- Yes, it is continuous there
- No, it jumps there
- No, it is undefined there
- Yes, and it is differentiable there
A subspace must be nonempty. Sam says the empty set already satisfies closure vacuously, so nonempty exists only to rule it out. Is Sam right, and why?
- Yes. Closure alone lets the empty set pass vacuously
- No. The empty set violates closure under addition
- No. Nonempty is decorative and changes nothing
f(x) equals x squared is continuous on the real line but not uniformly so. Which statement captures the difference between the two definitions?
- Uniform continuity lets the gap depend on the location
- Pointwise continuity already quantifies over all pairs at once
- Uniformity demands one tolerance working for every pair at once
For f(x) = x squared at x = 1, the secant slopes with step 0.5 and step -0.5 are 2.5 and 1.5. What is their average?
Answer: ______________

Answer key

For grown-ups. Fold this page away before handing over the rest.

Definitions are choices with a price W1-mt_b6ip8eP0-d-s1

False · The formulas agree where both are defined, but f is undefined at 0.
4 · The two formulas agree at -2, -1, 1, and 2, and differ only at 0, giving 4 integers.
False · The formulas agree wherever both are defined, but x squared over x is undefined at 0 while x is defined there.
Deciding whether two formulas define the same function · As a set of pairs, a function is pinned down by its graph set.
2 · f(0) = 1 and f(1/2) = 1 are rational points, while f(sqrt(2)) = 0, totalling 2.
True · Closure alone lets the empty set pass vacuously, so nonempty is the clause that keeps the definition honest.
Yes, it is continuous there · Thomae's function is continuous at every irrational, so at sqrt(2)/2 it is continuous there, though not differentiable.
Yes. Closure alone lets the empty set pass vacuously · With no members to test, every universal closure claim holds vacuously.
Uniformity demands one tolerance working for every pair at once · The single global tolerance is what fails for x squared on the whole line.
2 · The two one-sided secant slopes are 2.5 and 1.5, and their average is 2, the derivative.