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What Is a Function?: 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 Matching input and output

For $f(x)=x^2+x$ on all real numbers, find every input for which the output equals the input.
1. Hint 1
  The quantity being compared with the output is the input itself.
2. Hint 2
  Write $f(x)=x$ and remove the same linear term from both sides.
Answer

$x=0$ .

Full solution

Equality of the output and input means
$x^2+x=x.$
Subtracting $x$ from both sides gives
$x^2=0.$
The only real number with square zero is $0$ , so $x=0$ .
Checking gives $f(0)=0$ , which equals the input.

Answer

$x=0$ .

Key idea

An input that is unchanged by a function satisfies an equation equating the rule to that input.
Problem 1: Mark complete
Problem 2 An expression input

Let $f(x)=\frac{2x-1}{x+1}$ for $x\ne-1$ . Write $f(a-1)$ as a single fraction, and state the condition on $a$ that it needs.
1. Hint 1
  An expression in the input position replaces the entire input in each place.
2. Hint 2
  Substitute into both the numerator and denominator, then check when the new denominator is zero.
Answer

$\frac{2a-3}{a}$ , with $a\ne0$ .

Full solution

Replace every $x$ with $a-1$ .
$f(a-1)=\frac{2(a-1)-1}{(a-1)+1}.$
Simplifying gives
$f(a-1)=\frac{2a-3}{a}.$

The input $a-1$ must not equal $-1$ , so $a\ne0$ .
This agrees with the denominator of the resulting fraction.

Answer

$\frac{2a-3}{a}$ , with $a\ne0$ .

Key idea

An expression placed in the input slot must itself avoid the rule’s excluded inputs.
Problem 2: Mark complete
Problem 3 A table of outputs

Let $f(x)=x^2-2x$ , with domain exactly $\{-2,-1,1,3\}$ . Pair each allowed input with its output in a table, and state the range of the function.
1. Hint 1
  Evaluating the rule at an input means substituting that number for $x$ everywhere it appears.
2. Hint 2
  Take the inputs one at a time, and keep parentheses around a negative input so that its square comes out positive.
3. Hint 3
  The range collects the values the rule actually produces, each value listed once.
Answer

$f(-2)=8$ , $f(-1)=3$ , $f(1)=-1$ , $f(3)=3$ ; range $\{-1,3,8\}$ .

Full solution

Substitute each allowed input for $x$ in $x^2-2x$ .
At $x=-2$ the rule gives
$f(-2)=(-2)^2-2(-2),$
which simplifies to $8$ .
At $x=-1$ the rule gives
$f(-1)=(-1)^2-2(-1),$
which simplifies to $3$ .
At $x=1$ the rule gives
$f(1)=1^2-2(1),$
which simplifies to $-1$ .
At $x=3$ the rule gives
$f(3)=3^2-2(3),$
which simplifies to $3$ .

The four pairs of the table are therefore
$\begin{aligned}f(-2)&=8\\f(-1)&=3\\f(1)&=-1\\f(3)&=3\end{aligned}$

The range is the set of outputs actually produced, so it is $\{-1,3,8\}$ .
The value $3$ comes from two different inputs, which a function allows, and the range lists it once.

Answer

$f(-2)=8$ , $f(-1)=3$ , $f(1)=-1$ , $f(3)=3$ ; range $\{-1,3,8\}$ .

Key idea

A rule and a table of its input-output pairs show the same function, and its range lists each produced output once.
Problem 3: Mark complete
Problem 4 A distance rule on four inputs

A rule returns the distance between its input and $2$ on a number line. Its domain is exactly $\{-2,0,1,3\}$ . Write the rule in function notation, give its range, and plot all its pairs on the blank grid.

A blank coordinate grid.

Text description of this figure

A blank coordinate grid. The horizontal x-axis runs from negative three to four and the vertical y-axis from negative one to four, with gridlines, tick marks and number labels at every whole number, equal unit lengths on both axes, arrowheads at both ends of each axis, and the origin labeled 0. Nothing is plotted: there are no points, curves, coordinate labels or guide lines anywhere on the grid.
1. Hint 1
  Distance is the absolute value of the gap between two numbers.
2. Hint 2
  Evaluate the rule at every allowed input; the outputs give both the plotted heights and the range.
3. Hint 3
  Plot separate points because the rule allows exactly the four stated inputs.
Answer

Any letter may name the rule, for example $f(x)=|x-2|$ , equivalently $|2-x|$ ; range $\{1,2,4\}$ ; points $(-2,4)$ , $(0,2)$ , $(1,1)$ , $(3,1)$ .

Full solution

The distance rule is $f(x)=|x-2|$ , which may equally be written $|2-x|$ .
Its four outputs are
$\begin{aligned}f(-2)&=4\\f(0)&=2\\f(1)&=1\\f(3)&=1\end{aligned}$

The range is therefore $\{1,2,4\}$ , the three attained values.
The inputs $1$ and $3$ sit the same distance from $2$ , so they share the output $1$ .
Plot $(-2,4)$ , $(0,2)$ , $(1,1)$ , and $(3,1)$ , with no connecting segments.
Each point’s height agrees with the stated distance.

Answer

Any letter may name the rule, for example $f(x)=|x-2|$ , equivalently $|2-x|$ ; range $\{1,2,4\}$ ; points $(-2,4)$ , $(0,2)$ , $(1,1)$ , $(3,1)$ .

Key idea

Words, a rule, and plotted input-output pairs can describe the same function with a specified domain.
Problem 4: Mark complete
Problem 5 The real outputs

For $g(x)=\sqrt{10-2x}$ , allow every real input that produces a real output. State its domain and range.
1. Hint 1
  A real square root needs a quantity that is zero or positive inside it.
2. Hint 2
  Solve the condition on the input, and consider the possible nonnegative outputs of the root.
3. Hint 3
  To check the whole proposed range, solve for an input that produces a chosen output.
Answer

Domain: $x\le5$ . Range: $y\ge0$ .

Full solution

The root is defined when
$10-2x\ge0.$
Subtract $10$ and divide by $-2$ , reversing the inequality, to obtain
$x\le5.$

A square root produces no negative output.
To reach any chosen $r\ge0$ , solve $10-2x=r^2$ , giving
$x=\frac{10-r^2}{2}.$
This input is at most $5$ and produces $\sqrt{r^2}=r$ , so every nonnegative output is attained.

Answer

Domain: $x\le5$ . Range: $y\ge0$ .

Key idea

A complete range description both rules out impossible outputs and shows how every claimed output is reached.
Problem 5: Mark complete
Problem 6 The plotted relation

The graph contains every pair in a relation. Find all inputs that give output $2$ , and decide whether the entire relation is a function.

The graph of the relation.

Text description of this figure

A coordinate grid. The horizontal x-axis runs from negative three to five and the vertical y-axis from negative one to four, with gridlines, tick marks and number labels at every whole number, equal unit lengths on both axes, arrowheads at both ends of each axis, and the origin labeled 0. Two solid segments are drawn end to end. The first falls from the point negative two, three to the point one, zero. The second rises from the point one, zero to the point four, three. A filled dot marks each of those three points. These segments are the whole graph, and no point carries its coordinates.
1. Hint 1
  A point’s height is the output there, and its horizontal position is the input.
2. Hint 2
  Locate every point at height $2$ ; then, for the one-output condition, look along each vertical line and check whether any input has more than one plotted height.
Answer

Inputs $-1$ and $3$ ; yes, the relation is a function.

Full solution

The left segment runs from $(-2,3)$ to $(1,0)$ and has rule $y=1-x$ .
At height $2$ ,
$1-x=2,$
so $x=-1$ .
The right segment has rule $y=x-1$ , and
$x-1=2$
gives $x=3$ .

Every vertical line through the horizontal span meets the graph once.
The segments meet at the same point $(1,0)$ , so the shared endpoint does not give two outputs.
The relation is a function even though two inputs produce output $2$ .

Answer

Inputs $-1$ and $3$ ; yes, the relation is a function.

Key idea

Several inputs may share an output while every input still has exactly one output.
Problem 6: Mark complete
Problem 7 The strip lengths

A machine makes a strip whose length in centimeters is $L(n)=3n+2$ . Its setting $n$ must be a whole number from $0$ through $4$ . State the domain and range of this model, and explain why no setting produces a strip $10$ centimeters long.
1. Hint 1
  The setting restrictions determine which inputs the model permits.
2. Hint 2
  Compute the output for each allowed setting, then compare the requested length with the list of lengths you obtain.
Answer

Domain: $\{0,1,2,3,4\}$ . Range: $\{2,5,8,11,14\}$ centimeters. A length of $10$ centimeters would need $n=\tfrac83$ , which is not a whole number from $0$ through $4$ .

Full solution

The domain is the five whole-number settings.
The resulting lengths are
$\begin{aligned}L(0)&=2\\L(1)&=5\\L(2)&=8\\L(3)&=11\\L(4)&=14\end{aligned}$

These values give the range in centimeters, and $10$ is not among them.
Solving
$3n+2=10$
gives $n=\tfrac83$ , which is not one of the five whole-number settings, so no allowed setting produces that length.

Answer

Domain: $\{0,1,2,3,4\}$ . Range: $\{2,5,8,11,14\}$ centimeters. A length of $10$ centimeters would need $n=\tfrac83$ , which is not a whole number from $0$ through $4$ .

Key idea

A model’s stated input choices may give a finite domain and a finite range even when its formula accepts other numbers.
Problem 7: Mark complete
Problem 8 Two pieces of a graph

Each of the two drawn segments is the graph of a function on its own horizontal span. Mina claims their combined graph must also be a function. Is her claim correct? Identify an input that settles the question.

The two drawn segments.

Text description of this figure

A coordinate grid. The horizontal x-axis runs from negative four to four and the vertical y-axis from negative one to five, with gridlines, tick marks and number labels at every whole number, equal unit lengths on both axes, arrowheads at both ends of each axis, and the origin labeled 0. Two solid segments are drawn, and they are not joined to each other. The first rises from the point negative three, zero to the point one, four. The second rises from the point one, zero to the point three, two. A filled dot marks each of these four endpoints, and no endpoint carries its coordinates.
1. Hint 1
  Combining two valid graphs can create a conflict where their inputs overlap.
2. Hint 2
  Check the horizontal position shared by an endpoint of each segment.
Answer

No; input $1$ has outputs $4$ and $0$ .

Full solution

The first segment includes $(1,4)$ , while the second includes $(1,0)$ .
Thus the combined graph pairs the same input with two distinct outputs:
$4\ne0.$
A vertical line at $x=1$ meets both endpoints, so Mina’s claim is false.

Answer

No; input $1$ has outputs $4$ and $0$ .

Key idea

Combining functions on overlapping domains requires their outputs to agree at every shared input.
Problem 8: Mark complete
Problem 9 A small function

Create a function whose domain is exactly $\{0,1,2\}$ and whose range is exactly $\{7,9\}$ . Give all its ordered pairs, and explain why at least one output must be used by more than one input.
1. Hint 1
  Every domain value needs one output, and both named range values must appear.
2. Hint 2
  Try giving the three inputs three different outputs, and see what goes wrong with the stated range.
Answer

For example, $\{(0,7),(1,9),(2,7)\}$ ; at least one output is repeated.

Full solution

One construction assigns $0$ to $7$ , $1$ to $9$ , and $2$ to $7$ .
Each input has exactly one output, and the attained output values are exactly $7$ and $9$ .

If no output were repeated, the two outputs could serve at most two inputs.
The domain has three inputs, so at least one output must occur more than once.
Other assignments satisfying both requirements are also valid.

Answer

For example, $\{(0,7),(1,9),(2,7)\}$ ; at least one output is repeated.

Key idea

A function may need repeated outputs when its domain has more values than its range.
Problem 9: Mark complete
Problem 10 A machine adjustment

A machine returns $x/x$ for every nonzero real input $x$ . It is adjusted to return $5$ when its input is $0$ , while keeping the old outputs for every nonzero input. Lee says the adjusted machine is a function on all real numbers. Is Lee correct? State the adjusted machine’s range and explain.
1. Hint 1
  Check the output assigned to zero separately from the outputs assigned to nonzero inputs.
2. Hint 2
  A function requires one output per allowed input, not the same formula at every input.
Answer

Yes; the range is $\{1,5\}$ .

Full solution

For a nonzero input, the output is
$\frac{x}{x}=1.$
The adjustment separately assigns output $5$ to input $0$ .
It does not attempt to divide zero by zero.

Every real input now has exactly one assigned output.
Nonzero inputs produce $1$ , and zero produces $5$ , so the range is exactly $\{1,5\}$ and Lee is correct.

Answer

Yes; the range is $\{1,5\}$ .

Key idea

A rule can assign a missing input its own output while retaining exactly one output for every allowed input.
Problem 10: Mark complete

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