Algebra 1 Functions Worksheet Answer Key

Contents

This algebra 1 functions worksheet answer key solves all 23 problems of the functions, domain, range and function notation worksheet step by step.

Every solution shows the work, with the final answer in bold. Compare your reasoning, not only your result: for graphs, check that you drew the vertical lines, and for stories, check that your answer includes units.

Algebra 1Common Core: F-IF.A.1 · F-IF.A.2 · F-IF.B.5

Algebra 1 Functions Worksheet Answer Key: problems 1–5

Problem 1: Function from ordered pairs

List the inputs: 1, 2, 3, 4. No input is repeated, so each input has exactly one output.

Yes, it is a function.

Problem 2: Repeated input

The input 2 appears twice: it is paired with 5 and also with 9. One input has two outputs.

No, it is not a function.

Problem 3: Mapping diagram

Inputs 1, 2, 3, 4 each have exactly one arrow. The outputs 5, 5, 7, 9 repeat the 5, which is allowed.

Yes, it is a function.

Problem 4: Function from a table

The input $x=2$ appears twice, with outputs 4 and 6.

No, $y$ is not a function of $x$.

Problem 5: Vertical line test: curve

Draw the vertical line $x=1$. The curve satisfies $1=y^{2}-3$, so $y^{2}=4$ and $y=2$ or $y=-2$. The line meets the graph twice.

The vertical line $x=1$ meets the curve at $(1,2)$ and $(1,-2)$.

Not a function.

Algebra 1 Functions Worksheet Answer Key: problems 6–10

Problem 6: Vertical line test: V shape

Every vertical line meets the V-shaped graph in exactly one point.

Each vertical line meets the V-shaped graph exactly once.

It is a function.

Problem 7: Evaluate a linear function

$f(4)=3(4)-5=12-5=7$

$f(-2)=3(-2)-5=-6-5=-11$

$f(0)=3(0)-5=-5$

$f(4)=7$, $f(-2)=-11$, $f(0)=-5$

Problem 8: Evaluate a quadratic function

$g(3)=9+6-8=7$

$g(-4)=(-4)^{2}+2(-4)-8=16-8-8=0$

$g(0)=0+0-8=-8$

$g(3)=7$, $g(-4)=0$, $g(0)=-8$

Problem 9: Evaluate an absolute value function

$h(1)=|1-4|+3=|-3|+3=3+3=6$

$h(4)=|0|+3=3$

$h(9)=|5|+3=8$

$h(1)=6$, $h(4)=3$, $h(9)=8$

Problem 10: Negative and fractional inputs

$k(-3)=-2(9)+5(-3)=-18-15=-33$. Square first: $(-3)^{2}=9$.

$k(\frac{1}{2})=-2\cdot\frac{1}{4}+\frac{5}{2}=-\frac{1}{2}+\frac{5}{2}=2$

$k(-3)=-33$ and $k(\frac{1}{2})=2$

Algebra 1 Functions Worksheet Answer Key: problems 11–15

Problem 11: Find the input

$2x+7=19$, so $2x=12$ and $x=6$. Check: $2(6)+7=19$.

$2x+7=-5$, so $2x=-12$ and $x=-6$. Check: $2(-6)+7=-5$.

$x=6$ and $x=-6$

Problem 12: Read a function table

Find the column with $x=2$: the output is 0, so $f(2)=0$.

The column $x=-1$ has output 3, so $f(-1)=3$.

Outputs equal to 3 appear for $x=-1$ and $x=3$.

$f(2)=0$, $f(-1)=3$, and $x=-1$ or $x=3$

Problem 13: Read a function graph

At $x=0$ the graph is at $y=-3$. At $x=4$ it is at $y=5$. The graph touches the $x$-axis at $x=-1$ and $x=3$.

The graph of $f(x)=x^{2}-2x-3$.

$f(0)=-3$, $f(4)=5$, and $x=-1$ or $x=3$

Problem 14: Input with a variable

$f(a+1)=3(a+1)-2=3a+3-2=3a+1$

$f(2a)=3(2a)-2=6a-2$

$f(a+1)=3a+1$ and $f(2a)=6a-2$

Problem 15: Domain and range of a segment

The graph runs from $x=-2$ to $x=4$, and both endpoints are closed. It runs from $y=1$ to $y=5$.

Domain $-2\le x\le 4$ (blue). Range $1\le y\le 5$ (orange).

Domain: $-2\le x\le 4$. Range: $1\le y\le 5$.

Algebra 1 Functions Worksheet Answer Key: problems 16–20

Problem 16: Domain and range of a parabola

The graph extends left and right without end, so every real number is an input. The highest point is the vertex $(-1,3)$ and the graph goes down forever.

The highest point is $(-1,3)$, so every $y$-value is 3 or less.

Domain: all real numbers. Range: $y\le 3$.

Problem 17: Cupcake sale (discrete)

Only whole numbers of cupcakes make sense: the domain is $\{1,2,3,4,5,6\}$. Then $C(n)=3n$ gives $3, 6, 9, 12, 15, 18$.

Domain: $\{1,2,3,4,5,6\}$. Range: $\{3,6,9,12,15,18\}$ dollars. Discrete.

Problem 18: Birthday relation

Each student has exactly one birthday, so every input (student) has one output (date): students to birthdays is a function.

Two students in the same class can share a birthday. Then one date would be paired with two students, so birthdays to students is not a function.

Problem 19: Plumber charges

The input 3 is hours and the output 115 is dollars. So a 3-hour job costs $\$115$. Check: $25(3)+40=115$.

$C(2.5)=25(2.5)+40=62.5+40=102.5$

A 3-hour job costs $\$115$; $C(2.5)=\$102.50$.

Problem 20: Height of a tossed ball

Check: $h(2)=-16(4)+48(2)+4=-64+96+4=36$. It means that 2 seconds after the toss, the ball is 36 feet high.

$h(1.5)=-16(2.25)+48(1.5)+4=-36+72+4=40$

At 2 seconds the ball is 36 ft high; $h(1.5)=40$ ft.

Algebra 1 Functions Worksheet Answer Key: problems 21–23

Problem 21: Filling a tank (continuous)

The tank is full when $5t=40$, so $t=8$ minutes. Time can be any value from 0 to 8, and the volume can be any value from 0 to 40.

Gallons in the tank after $t$ minutes: a connected segment.

Domain: $0\le t\le 8$. Range: $0\le V\le 40$. Continuous.

Problem 22: Taxi fare

$F(6)=2.50+1.75(6)=2.50+10.50=13$, so a 6-mile ride costs $\$13.00$.

Solve $2.50+1.75m=20$: $1.75m=17.50$, so $m=10$.

$F(6)=\$13.00$; a $\$20.00$ ride is 10 miles.

Problem 23: Garden area

$A(8)=8(12)=96$.

Test widths: $A(5)=5(15)=75$ and $A(15)=15(5)=75$, so $w=5$ or $w=15$.

Both width and length must be positive, so $0 < w < 20$. The area is largest at $w=10$: $A(10)=100$. The area is always positive, so $0 < A\le 100$.

Area of the garden for each width. The highest point is $(10,100)$.

$A(8)=96$ ft$^{2}$; $w=5$ or $15$ ft; domain $0 < w < 20$; range $0 < A\le 100$.

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