Algebra 1 Building Worksheet Answer Key – Free PDF

Contents

This algebra 1 building worksheet answer key solves all 23 problems of the building functions and inverses worksheet step by step.

Every solution shows the setup, the arithmetic and a check, with the final answer in bold. Compare your steps with these, especially the order of operations when you write a function and the minus sign when you subtract a cost function.

Algebra 1Common Core: F-BF.A.1a · F-BF.A.1b · F-BF.B.4a · F-LE.A.1b

Algebra 1 Building Worksheet Answer Key: problems 1–5

Problem 1: Steps to a function

Multiplying by 4 gives $4x$ and subtracting 7 gives $f(x)=4x-7$.

$f(5)=4(5)-7=20-7=13$. $f(x)=4x-7$, $f(5)=13$

Problem 2: Recursive growth

Start: $h(0)=6$. Each week adds 2: $h(n)=h(n-1)+2$.

$h(3)=10+2=12$ and $h(4)=12+2=14$. $h(4)=14$ inches

Problem 3: Recursive to explicit

The value starts at 5 and 3 is added $n$ times , so $f(n)=3n+5$.

$f(10)=3(10)+5=35$. $f(n)=3n+5$, $f(10)=35$

Problem 4: Order of the steps

Add 6 first, so the sum must be in parentheses: $g(x)=3(x+6)$.

$g(4)=3(4+6)=3(10)=30$. $g(x)=3(x+6)$, $g(4)=30$

Problem 5: Taxi fare function

Cost is the flat fee plus the rate times the miles: $c(m)=2.25m+3$.

$c(8)=2.25(8)+3=18+3=21$. $c(m)=2.25m+3$; the ride costs $21.00

Algebra 1 Building Worksheet Answer Key: problems 6–10

Problem 6: Sum of two functions

$(f+g)(x)=(3x+2)+(x-5)=4x-3$.

$(f+g)(2)=4(2)-3=5$. Check: $f(2)=8$ and $g(2)=-3$, and $8+(-3)=5$. $4x-3$; $5$

Problem 7: Difference of two functions

$(f-g)(x)=(5x-1)-(2x+8)=5x-1-2x-8=3x-9$.

$(f-g)(6)=3(6)-9=9$. Check: $f(6)=29$, $g(6)=20$, and $29-20=9$. $3x-9$; $9$

Problem 8: Product of two functions

$(f\cdot g)(x)=(x+3)(2x)=2x^{2}+6x$.

$(f\cdot g)(4)=2(16)+6(4)=32+24=56$. Check: $f(4)\cdot g(4)=7\cdot 8=56$. $2x^{2}+6x$; $56$

Problem 9: Profit function

$P(x)=R(x)-C(x)=6x-(2x+40)=4x-40$.

$P(25)=4(25)-40=60$. Check: $R(25)=150$, $C(25)=90$, $150-90=60$. $P(x)=4x-40$; profit $\$60$

Problem 10: Break-even from a graph

Break even when revenue equals cost: $7x=3.5x+84$, so $3.5x=84$ and $x=24$.

$R(24)=168$ and $C(24)=3.5(24)+84=84+84=168$. 24 hats, revenue $\$168$

The lines cross at (24, 168).

Algebra 1 Building Worksheet Answer Key: problems 11–15

Problem 11: Solve f(x) = c

$3x-4=20$, so $3x=24$ and $x=8$. Check: $f(8)=24-4=20$. $x=8$

Problem 12: Inverse of a linear function

$y=5x+10$, so $y-10=5x$ and $x=\frac{y-10}{5}=\frac{y}{5}-2$.

Check: $f(3)=25$ and $\frac{25-10}{5}=3$. $x=\frac{y-10}{5}$

Problem 13: Inverse with a fraction

$y=\frac{x}{4}+6$, so $y-6=\frac{x}{4}$ and $x=4(y-6)=4y-24$. The inverse undoes “add 6” first, then “divide by 4”.

For $y=9$: $x=4(9-6)=12$. Check: $f(12)=3+6=9$. $x=4y-24$; input $12$

Forward and inverse machines for this function.

Problem 14: Check an inverse

$f(5)=8$ and $g(8)=\frac{15}{3}=5$. Also $g(5)=\frac{12}{3}=4$ and $f(4)=12-7=5$.

Each function returns the original input, so $g$ is the inverse of $f$.

Problem 15: Is the rate constant?

$x$ rises by 2 each step and $y$ rises by 6 each step ($11-5=6$, $17-11=6$, $23-17=6$), so the rate is constant: $\frac{6}{2}=3$ per 1 unit.

The start value at $x=0$ is 5, so $f(x)=3x+5$. Check: $f(6)=23$. constant rate 3; $f(x)=3x+5$

Equal steps in x give equal changes in y.

Algebra 1 Building Worksheet Answer Key: problems 16–20

Problem 16: Compare two tables

Table A: $12-7=5$, $17-12=5$, $22-17=5$, always +5 for each 1 step in $x$.

Table B: $6-3=3$, $12-6=6$, $24-12=12$, the changes keep doubling. Only Table A has a constant rate (5 per unit)

Table A: constant +5.

Problem 17: Temperature conversion

(a) $1.8c+32=68$, so $1.8c=36$ and $c=20$.

(b) $c=\frac{F-32}{1.8}$. Check with 95°F: $\frac{95-32}{1.8}=\frac{63}{1.8}=35$ and $1.8(35)+32=95$. 20°C; $c=\frac{F-32}{1.8}$

Problem 18: Play ticket profit

$R(x)=8x$, $C(x)=1.25x+150$, $P(x)=8x-(1.25x+150)=6.75x-150$.

A profit needs $6.75x>150$, so $x$ is greater than 22.2. Test the whole numbers 22 and 23: $P(22)=148.5-150=-1.5$ (a loss) and $P(23)=155.25-150=5.25$. 23 tickets

Problem 19: Taxi fare inverse

$c-3=2.25m$, so $m=\frac{c-3}{2.25}$.

$m=\frac{25.5-3}{2.25}=\frac{22.5}{2.25}=10$. Check: $2.25(10)+3=25.5$. 10 miles

Problem 20: Savings recursion

Recursive: $S(0)=45$, $S(n)=S(n-1)+12$. Explicit: $S(n)=12n+45$.

$12n+45=165$, so $12n=120$ and $n=10$. Check: $12(10)+45=165$. 10 weeks

Algebra 1 Building Worksheet Answer Key: problems 21–23

Problem 21: Hot dog stand profit

$P(x)=3.5x-(1.1x+48)=2.4x-48$.

Break even: $2.4x=48$, so $x=20$. Profit for 50: $P(50)=2.4(50)-48=120-48=72$. 20 hot dogs; profit $\$72$

Break-even at (20, 70).

Problem 22: Gas tank inverse

$0.04m=14-g$, so $m=\frac{14-g}{0.04}=350-25g$.

For $g=6$: $m=350-150=200$. For $g=0$: $m=350$. Check: $14-0.04(200)=6$. 200 miles; 350 miles

Problem 23: Two classes combined

(a) $T(x)=(4x-20)+(6x-35)=10x-55$.

(b) $10x-55=145$, so $10x=200$ and $x=20$. Check: $L(20)=60$, $C(20)=85$, $60+85=145$.

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Read this algebra 1 building worksheet answer key after trying each problem yourself, and redo any problem you missed.

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Algebra 1 Building Functions and Inverses worksheet answer key

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