Lesson test: Functions

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Functions

1.[1p]
The functions $s : ℝ \to ℝ$ and $t : ℝ \to [0, \infty)$ , both given by $x \mapsto x^{2}$ , are the same function, since they have the same value at every real number.

True

The answer is: False

False

Correct

2.[1p]
Let $s : ℝ \to ℝ$ , $s (x) = x^{2}$ . What is the preimage $s^{- 1} ([1, 4])$ ?

$[- 2, - 1] \cup [1, 2]$

Correct

$[1, 2]$

The answer is: $[- 2, - 1] \cup [1, 2]$

$[1, 16]$

The answer is: $[- 2, - 1] \cup [1, 2]$

It does not exist, because $s$ has no inverse

The answer is: $[- 2, - 1] \cup [1, 2]$

3.[3p]
Let $f : A \to B$ be any function, $S \subseteq A$ and $T \subseteq B$ . Which statements are always true?

Select all that apply

$f (f^{- 1} (T)) \subseteq T$

Correct

$S \subseteq f^{- 1} (f (S))$

Correct

$T \subseteq f (f^{- 1} (T))$

The answer is: $f (f^{- 1} (T)) \subseteq T$ , $S \subseteq f^{- 1} (f (S))$ , If $f$ is surjective, then $f (f^{- 1} (T)) = T$

$f^{- 1} (f (S)) \subseteq S$

The answer is: $f (f^{- 1} (T)) \subseteq T$ , $S \subseteq f^{- 1} (f (S))$ , If $f$ is surjective, then $f (f^{- 1} (T)) = T$

If $f$ is surjective, then $f (f^{- 1} (T)) = T$

Correct

4.[3p]
Match each function to its properties.

$x \mapsto 3 x + 7$ from $ℝ$ to $ℝ$
$n \mapsto 2 n$ from $ℕ$ to $ℕ$
$x \mapsto x^{2}$ from $ℝ$ to $[0, \infty)$
$x \mapsto x^{2}$ from $ℝ$ to $ℝ$

neither injective nor surjective
bijective
injective, not surjective
surjective, not injective

Show the answer

$x \mapsto 3 x + 7$ from $ℝ$ to $ℝ$ : bijective $n \mapsto 2 n$ from $ℕ$ to $ℕ$ : injective, not surjective $x \mapsto x^{2}$ from $ℝ$ to $[0, \infty)$ : surjective, not injective $x \mapsto x^{2}$ from $ℝ$ to $ℝ$ : neither injective nor surjective

5.[2p]
The proof that $x \mapsto 3 x + 7$ is surjective on $ℝ$ puts $a = \frac{b - 7}{3}$ . For the same rule from $ℤ$ to $ℤ$ , which step of that proof breaks down?

Checking that $a$ belongs to the domain, since $\frac{b - 7}{3}$ need not be an integer

Correct

Checking that $f (a) = b$ , since the algebra changes on the integers

The answer is: Checking that $a$ belongs to the domain, since $\frac{b - 7}{3}$ need not be an integer

Solving $f (a) = b$ in the scratch work, since integers cannot be divided

The answer is: Checking that $a$ belongs to the domain, since $\frac{b - 7}{3}$ need not be an integer

None: the rule is surjective on $ℤ$ as well

The answer is: Checking that $a$ belongs to the domain, since $\frac{b - 7}{3}$ need not be an integer

6.[2p]
Put the lines of the proof that if $f : A \to B$ , $g : B \to C$ and $g \circ f$ is surjective, then $g$ is surjective, in order.

Put $b = f (a)$ , which is an element of $B$ .
Since $c$ was arbitrary, $g$ is surjective.
Since $g \circ f$ is surjective, there is $a \in A$ with $g (f (a)) = c$ .
Let $c \in C$ .
Then $g (b) = g (f (a)) = c$ .

Show the answer

a, b, c, d, e

7.[2p]
Functions $f : A \to B$ and $g : B \to A$ satisfy $g \circ f = {id}_{A}$ , and nothing more is known. What follows?

$f$ is injective

Correct

$f$ is surjective

The answer is: $f$ is injective

$f$ is a bijection and $g$ is its inverse

The answer is: $f$ is injective

$g$ is injective

The answer is: $f$ is injective

8.[1p]
How many functions are there from a set with 3 elements to a set with 5 elements?

CorrectNot quite: 125

9.[2p]
How many injective functions are there from a set with 3 elements to a set with 5 elements?

CorrectNot quite: 60

10.[3p]
Let $f : A \to B$ be any function, $S_{1}, S_{2} \subseteq A$ and $T_{1}, T_{2}, T \subseteq B$ . Which equalities always hold?

Select all that apply

$f^{- 1} (T_{1} \cap T_{2}) = f^{- 1} (T_{1}) \cap f^{- 1} (T_{2})$

Correct

$f^{- 1} (B ∖ T) = A ∖ f^{- 1} (T)$

Correct

$f (S_{1} \cup S_{2}) = f (S_{1}) \cup f (S_{2})$

Correct

$f (S_{1} \cap S_{2}) = f (S_{1}) \cap f (S_{2})$

The answer is: $f^{- 1} (T_{1} \cap T_{2}) = f^{- 1} (T_{1}) \cap f^{- 1} (T_{2})$ , $f^{- 1} (B ∖ T) = A ∖ f^{- 1} (T)$ , $f (S_{1} \cup S_{2}) = f (S_{1}) \cup f (S_{2})$

$f (A ∖ S_{1}) = B ∖ f (S_{1})$

The answer is: $f^{- 1} (T_{1} \cap T_{2}) = f^{- 1} (T_{1}) \cap f^{- 1} (T_{2})$ , $f^{- 1} (B ∖ T) = A ∖ f^{- 1} (T)$ , $f (S_{1} \cup S_{2}) = f (S_{1}) \cup f (S_{2})$

11.[2p]
A function $f : A \to B$ has a right inverse, a function $g : B \to A$ with $f \circ g = {id}_{B}$ , exactly when:

$f$ is surjective

Correct

$f$ is injective

The answer is: $f$ is surjective

$f$ is a bijection

The answer is: $f$ is surjective

$A$ is not empty

The answer is: $f$ is surjective

12.[2p]
Let $f : {1, 2} \to {a, b, c}$ be given by $f (1) = a$ and $f (2) = b$ . How many left inverses does $f$ have, that is, functions $g : {a, b, c} \to {1, 2}$ with $g \circ f = id$ ?

CorrectNot quite: 2

13.[1p]
Every injective function from $ℕ$ to $ℕ$ is surjective.

True

The answer is: False

False

Correct

1.[1p] The functions s:ℝ→ℝ and t:ℝ→[0,∞), both given by x↦x2, are the same function, since they have the same value at every real number.

2.[1p] Let s:ℝ→ℝ, s(x)=x2. What is the preimage s−1([1,4])?

3.[3p] Let f:A→B be any function, S⊆A and T⊆B. Which statements are always true?

4.[3p] Match each function to its properties.

5.[2p] The proof that x↦3x+7 is surjective on ℝ puts a=b−73. For the same rule from ℤ to ℤ, which step of that proof breaks down?

6.[2p] Put the lines of the proof that if f:A→B, g:B→C and g∘f is surjective, then g is surjective, in order.

7.[2p] Functions f:A→B and g:B→A satisfy g∘f=idA, and nothing more is known. What follows?

8.[1p] How many functions are there from a set with 3 elements to a set with 5 elements?

9.[2p] How many injective functions are there from a set with 3 elements to a set with 5 elements?

10.[3p] Let f:A→B be any function, S1,S2⊆A and T1,T2,T⊆B. Which equalities always hold?

11.[2p] A function f:A→B has a right inverse, a function g:B→A with f∘g=idB, exactly when:

12.[2p] Let f:{1,2}→{a,b,c} be given by f(1)=a and f(2)=b. How many left inverses does f have, that is, functions g:{a,b,c}→{1,2} with g∘f=id?

13.[1p] Every injective function from ℕ to ℕ is surjective.

1.[1p]
The functions $s : ℝ \to ℝ$ and $t : ℝ \to [0, \infty)$ , both given by $x \mapsto x^{2}$ , are the same function, since they have the same value at every real number.

2.[1p]
Let $s : ℝ \to ℝ$ , $s (x) = x^{2}$ . What is the preimage $s^{- 1} ([1, 4])$ ?

3.[3p]
Let $f : A \to B$ be any function, $S \subseteq A$ and $T \subseteq B$ . Which statements are always true?

4.[3p]
Match each function to its properties.

5.[2p]
The proof that $x \mapsto 3 x + 7$ is surjective on $ℝ$ puts $a = \frac{b - 7}{3}$ . For the same rule from $ℤ$ to $ℤ$ , which step of that proof breaks down?

6.[2p]
Put the lines of the proof that if $f : A \to B$ , $g : B \to C$ and $g \circ f$ is surjective, then $g$ is surjective, in order.

7.[2p]
Functions $f : A \to B$ and $g : B \to A$ satisfy $g \circ f = {id}_{A}$ , and nothing more is known. What follows?

8.[1p]
How many functions are there from a set with 3 elements to a set with 5 elements?

9.[2p]
How many injective functions are there from a set with 3 elements to a set with 5 elements?

10.[3p]
Let $f : A \to B$ be any function, $S_{1}, S_{2} \subseteq A$ and $T_{1}, T_{2}, T \subseteq B$ . Which equalities always hold?

11.[2p]
A function $f : A \to B$ has a right inverse, a function $g : B \to A$ with $f \circ g = {id}_{B}$ , exactly when:

12.[2p]
Let $f : {1, 2} \to {a, b, c}$ be given by $f (1) = a$ and $f (2) = b$ . How many left inverses does $f$ have, that is, functions $g : {a, b, c} \to {1, 2}$ with $g \circ f = id$ ?

13.[1p]
Every injective function from $ℕ$ to $ℕ$ is surjective.