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Relations and Functions: Core practice

10 practice problems for this lesson. Work on paper, use hints when you need them, and check the answer or the full solution when you are ready.

Difficulty: Core (core-course level)

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Problem 1 of 10

Problem 1 Three storage bins

A record pairs bin 1 with shelf 4, bin 2 with shelf 4, and bin 4 with shelf 1. Write this relation as a set of ordered pairs with the shelf number first.
1. Hint 1
  The order named in the question determines the order inside each pair.
2. Hint 2
  For each record, put the shelf number before its associated bin number.
Answer

$\{(4,1),(4,2),(1,4)\}$ .

Full solution

The first record becomes $(4,1)$ and the second becomes $(4,2)$ .
The last record becomes $(1,4)$ .
The relation is
$R=\{(4,1),(4,2),(1,4)\}.$

Each pair preserves its original bin and shelf association while placing the shelf first.

Answer

$\{(4,1),(4,2),(1,4)\}$ .

Key idea

An ordered pair records not just two values but their assigned roles.
Problem 1: Mark complete
Problem 2 A permitted input

Find the natural real domain of $f(x)=\sqrt{\lvert x\rvert-2}$ .
1. Hint 1
  The quantity inside a real square root must be nonnegative.
2. Hint 2
  Translate that condition into a minimum distance from zero.
Answer

$(-\infty,-2]\cup[2,\infty)$ , that is, $x\le-2$ or $x\ge2$ .

Full solution

The root condition is
$\lvert x\rvert-2\ge0.$

Thus $\lvert x\rvert\ge2$ , which means $x\le-2$ or $x\ge2$ .
At both endpoints the root is zero and is defined, so they are included.

Answer

$(-\infty,-2]\cup[2,\infty)$ , that is, $x\le-2$ or $x\ge2$ .

Key idea

A natural domain is found by applying the restrictions of the operations actually used.
Problem 2: Mark complete
Problem 3 Four recorded inputs

A function has domain $\{-3,0,2,5\}$ , codomain $\mathbb R$ , and rule $f(x)=\lvert x-1\rvert$ . Find its range.
1. Hint 1
  The range contains the outputs actually reached by the declared inputs.
2. Hint 2
  Evaluate the four distances from $1$ and remove repeated outputs.
Answer

$\{1,4\}$ .

Full solution

The four outputs are
$\begin{aligned}f(-3)&=4\\f(0)&=1\\f(2)&=1\\f(5)&=4\end{aligned}$

The distinct outputs are $1$ and $4$ .
They form the range even though the codomain contains every real number.

Answer

$\{1,4\}$ .

Key idea

A finite declared domain may produce a much smaller range than the formula suggests.
Problem 3: Mark complete
Problem 4 Two separate ramps

The figure shows an entire relation, including its endpoints. Decide whether it gives a function with $x$ as input and whether it gives a function with $y$ as input. For the reading with $x$ as input, give the domain and range.

The two ramps.

Text description of this figure

A grid with the horizontal axis labeled x running from -4 to 4 and the vertical axis labeled y running from 0 to 4, gridlines and number labels at every whole number, and the origin labeled 0. Two separate rising segments are drawn, each with filled endpoints: one from (-3, 1) to (-1, 3), and one from (1, 1) to (3, 3). No other point, guide line, or domain and range label is drawn.
1. Hint 1
  Fix one input coordinate and look for two different corresponding output coordinates.
2. Hint 2
  Compare vertical and horizontal lines through the two segments.
Answer

Function of $x$ , not of $y$ ; domain $[-3,-1]\cup[1,3]$ ; range $[1,3]$ .

Full solution

The segments occupy disjoint horizontal intervals, and neither is vertical.
Thus every permitted $x$ has one $y$ .
The x-domain is
$[-3,-1]\cup[1,3].$

Both segments reach every height from $1$ through $3$ , so the range is $[1,3]$ .
A horizontal line such as $y=2$ meets the relation at $x=-2$ and $x=2$ .
Reading $y$ as input therefore assigns two outputs to that input.

Answer

Function of $x$ , not of $y$ ; domain $[-3,-1]\cup[1,3]$ ; range $[1,3]$ .

Key idea

The input choice determines which line test applies to a relation.
Problem 4: Mark complete
Problem 5 A partial reading

A sensor uses $V(t)=\sqrt{12-3t}$ only for real times $1\le t\le3$ , measured in seconds. Find the natural domain of the formula, the declared domain of the sensor, and the range of the sensor output.
1. Hint 1
  The formula restrictions and the allowed observation times are separate pieces of information.
2. Hint 2
  Find the root condition first, then evaluate the extreme allowed times.
Answer

Natural domain $t\le4$ ; declared domain $[1,3]$ seconds; output range $\sqrt3\le V(t)\le3$ .

Full solution

The root requires $12-3t\ge0$ , giving $t\le4$ .
The sensor instead declares only
$1\le t\le3.$
Over that interval,
$3\le12-3t\le9.$

Taking the nonnegative root gives
$\sqrt3\le V(t)\le3.$
Every value in the interval is reached as the inside quantity runs through $[3,9]$ , including both ends.

Answer

Natural domain $t\le4$ ; declared domain $[1,3]$ seconds; output range $\sqrt3\le V(t)\le3$ .

Key idea

A declared domain can narrow both the allowable inputs and the resulting range.
Problem 5: Mark complete
Problem 6 A missing record

A relation contains $(-2,5)$ , $(1,7)$ , and $(4,5)$ . Add one ordered pair with first coordinate $1$ and second coordinate $c$ so that the result is still a function of the first coordinate. Find every possible $c$ and state whether the added pair changes the relation.
1. Hint 1
  A function gives one output to each allowed input.
2. Hint 2
  Compare the proposed pair with the existing pair whose input is $1$ .
Answer

$c=7$ ; the relation does not change.

Full solution

The existing record requires the output at input $1$ to be $7$ .
Thus the new pair must satisfy
$c=7.$

Adding $(1,7)$ adds a pair already present, and a set does not acquire a new member when the same member is listed again.
Any other $c$ would assign two outputs to input $1$ .

Answer

$c=7$ ; the relation does not change.

Key idea

Repeating an existing pair leaves a relation unchanged, while conflicting outputs violate the function rule.
Problem 6: Mark complete
Problem 7 A distance range

Let $f(x)=\lvert x-1\rvert$ have declared domain $0<x\le4$ and codomain $[0,4]$ . Find the range and identify which values in the codomain are not produced.
1. Hint 1
  Find the smallest and largest distances attainable from $1$ on the declared domain.
2. Hint 2
  The input $1$ is allowed, and the part $1\le x\le4$ already produces a full interval of distances.
Answer

Range $[0,3]$ ; unused codomain values $(3,4]$ .

Full solution

The input $x=1$ gives output $0$ .
For $1\le x\le4$ ,
$f(x)=x-1,$
so all outputs in $[0,3]$ occur.

For $0<x<1$ , the output is $1-x$ , which lies inside $(0,1)$ and adds nothing outside $[0,3]$ .
Therefore the range is $[0,3]$ , and exactly the codomain values $3<y\le4$ are unused.

Answer

Range $[0,3]$ ; unused codomain values $(3,4]$ .

Key idea

To find a range, establish both a bound and that every value inside it is attained.
Problem 7: Mark complete
Problem 8 Deleting an input

A student claims that deleting one input from a function domain must delete a value from its range. Decide whether this is true, using $f(x)=\lvert x\rvert$ on $\{-2,0,2\}$ as evidence.
1. Hint 1
  Different inputs may produce the same output.
2. Hint 2
  Consider removing one of the two inputs that have equal magnitude.
Answer

False; deleting $-2$ leaves the range $\{0,2\}$ unchanged.

Full solution

Before deletion,
$\begin{aligned}f(-2)&=2\\f(0)&=0\\f(2)&=2\end{aligned}$

After deleting $-2$ , the remaining input $2$ still produces output $2$ , and $0$ still produces $0$ .
The domain changed but the range did not, disproving the claim.

Answer

False; deleting $-2$ leaves the range $\{0,2\}$ unchanged.

Key idea

Removing an input removes its output from the range only if no remaining input produces that output.
Problem 8: Mark complete
Problem 9 An output label

A rule assigns $g(x)=\sqrt{x}$ on the domain $\{0,1,4,9\}$ and declares codomain $\{0,1,2\}$ . Is this a valid function with those data? If not, give the smallest codomain containing the declared one that makes it valid.
1. Hint 1
  Every output must lie in the declared codomain.
2. Hint 2
  Check all four allowed inputs, especially the largest one.
Answer

No; the smallest enlarged codomain is $\{0,1,2,3\}$ .

Full solution

The outputs are $0,1,2,3$ .
In particular,
$g(9)=3,$
which is absent from the proposed codomain.

Adding $3$ makes every output permitted.
Nothing else needs to be added, so $\{0,1,2,3\}$ is the smallest codomain containing the original declaration.

Answer

No; the smallest enlarged codomain is $\{0,1,2,3\}$ .

Key idea

A codomain is valid only when it contains every output of the rule on the declared domain.
Problem 9: Mark complete
Problem 10 An extra record

A relation consists of $(0,2)$ , $(2,4)$ , and $(5,7)$ . A student says adding $(7,0)$ preserves the function property under either choice of input coordinate. Is the claim correct? Explain.
1. Hint 1
  Test whether the new pair creates a repeated input with a different output under either reading.
2. Hint 2
  Compare its first coordinate with all existing first coordinates, then do the same for second coordinates.
Answer

Yes; the enlarged relation is a function under either input choice.

Full solution

The first coordinates become $\{0,2,5,7\}$ and the second coordinates become $\{0,2,4,7\}$ .
There are no repetitions in either coordinate list.

Thus with either coordinate chosen as input, every input occurs in exactly one pair and receives exactly one output.
The claim holds.

Answer

Yes; the enlarged relation is a function under either input choice.

Key idea

A relation with no repeated first or second coordinates is a function under either input choice.
Problem 10: Mark complete

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