Expressions & Properties
Simplify completely.
3x(x-5)-4(x+2)
3x^2-15x-4x-8
3x^2-19x-8
Distribute each factor, then combine like terms.
Solve algebraically. Show the distributive step.
4+2(3x-5)=5(x+2)
4+6x-10=5x+10
6x-6=5x+10
x=16
Given the function below, find f(−2).
f(x)=-4x+9
f(-2)=-4(-2)+9=17
Substitute −2 for x, then follow the order of operations.
Does this relation define a function? Explain, even though one output repeats.
{(−5, 2), (0, 6), (3, 2), (8, 9)}
Yes. Every input has exactly one output. The output 2 can be paired with more than one input.
A science center charges $28 per student plus a flat $185 workshop fee for a school visit. Write a function C(n) for the total cost for n students. Then find C(12).
C(n)=28n+185
C(12)=28(12)+185=521
The total cost for 12 students is $521. The flat fee is added once.
Write an expression in standard form with leading coefficient 5, degree 3, and constant term −9. Include a nonzero linear term.
5x^3+2x-9
One possible answer. The highest-degree term is 5x³ and the constant is −9. Other nonzero linear coefficients are acceptable.
Solve. Give the exact answer as a simplified fraction.
3/4(x+2/3)=1.8
x+2/3=1.8 xx 4/3=12/5
x=12/5-2/3=26/15
Given the function below, evaluate g(−3). Show your substitution.
g(x)=2x^2-7
g(-3)=2(-3)^2-7=18-7=11
Square the entire negative input before multiplying by 2.
Does this table represent a function? Identify the evidence.
| Input | Output |
|---|---|
| -4 | 1 |
| 2 | 7 |
| 5 | 9 |
| 2 | 11 |
No. The input 2 is paired with both 7 and 11. One input cannot have two different outputs in a function.
A school has $7,200 to buy cameras and tablets. Each camera costs $165; each tablet costs $405. After buying 24 cameras, what is the greatest number of tablets the school can buy? Show your model.
165(24)+405t <= 7200
3960+405t <= 7200
405t <= 3240
t <= 8
The school can buy at most 8 tablets. The number of tablets must be a whole number.
Name both properties used in the step shown. Do not solve the equation.
4x^2+7-6x=2(5x-3)
4x^2-6x+7=10x-6
Commutative property of addition: reorder +7 and −6x on the left.
Distributive property of multiplication over addition: multiply both terms inside parentheses by 2.
Solve and state how many solutions the equation has. Justify your answer.
5(2x-3)-4x=3(2x-4)
10x-15-4x=6x-12
6x-15=6x-12
-15=-12
No solution. The equation reduces to a false statement.
For f(x) = −3x + 8, find the outputs for x = −1, 1, and 3. Write the results as ordered pairs.
f(-1)=11
f(1)=5
f(3)=-1
Ordered pairs: (−1, 11), (1, 5), (3, −1).
State the domain and range of this relation. List each value only once.
{(−3, 8), (1, −2), (4, 8), (7, 5)}
Domain: {−3, 1, 4, 7}.
Range: {−2, 5, 8}.
The domain contains the inputs; the range contains the outputs. Do not repeat 8 in the range.
Eli earns $22 for each lawn he completely mows. He can mow at most 18 lawns this week. Let E(n) represent his earnings. Write the function and a reasonable domain. Is the domain discrete or continuous?
E(n)=22n
Domain: whole numbers {0, 1, 2, …, 18}. It is discrete because n counts completely mowed lawns; fractional lawns are excluded in this model.
A student simplifies the expression below to 2x² + 5x − 12. Identify the sign error and give the correct simplified expression.
2x(x+4)-3(x-4)
2x^2+8x-3x+12=2x^2+5x+12
The product (−3)(−4) is +12, not −12.
Solve algebraically. Give the exact answer.
3-2(4x-5)=5(1-x)
3-8x+10=5-5x
13-8x=5-5x
8=3x
x=8/3
A student evaluates h(−4) as 37. Identify the error and give the correct value.
h(x)=-2x^2+5
h(-4)=-2(-4)^2+5=-32+5=-27
The negative coefficient is outside the square. Only the input is squared: (−4)² = 16. Then multiply by −2.
A tank drains continuously for 8 minutes until empty. V(t) gives the water remaining after t minutes. State a reasonable domain and say whether it is discrete or continuous. Explain.
All real values of t from 0 through 8, inclusive: 0 ≤ t ≤ 8. The domain is continuous because elapsed time can include fractional minutes, such as 2.6 minutes.
Solve the formula for t. Assume r ≠ 0.
d=r(t-p)
d/r=t-p
t=d/r+p
Divide both sides by r, then add p to both sides. Do not divide p by r.
Simplify completely, then identify the leading coefficient and constant term.
4x(2x-3)-2(3x^2-5x+7)
8x^2-12x-6x^2+10x-14
2x^2-2x-14
Leading coefficient: 2. Constant term: −14.
Solve algebraically. Show how you handle the decimal coefficients.
2750+1.06x=1900+1.11x
850=0.05x
x=850/0.05=17000
Subtract 1900 and 1.06x from both sides, then divide by 0.05. Check: both original sides equal 20,770.
The graphs of f and g intersect where their outputs are equal. Find every x-value for which f(x) = g(x), and check your answer.
f(x)=2x+9
g(x)=abs(x)
The only solution is x = −3.
If x ≥ 0: 2x + 9 = x gives x = −9, which does not satisfy x ≥ 0.
If x < 0: 2x + 9 = −x gives x = −3.
f(-3)=3=g(-3)
Choose the value of k that makes this relation a function. Explain why the other choices fail.
{(−2, 5), (1, 7), (4, 5), (k, 9)}
Choices: −2, 1, 4, or 6.
k = 6. This creates a new input with one output. Each other choice repeats an existing input but assigns it a different output. The repeated output 5 does not prevent the relation from being a function.
The area of a trapezoid is given below. Solve for b₂. Then find b₂ when A = 96 cm², h = 8 cm, and b₁ = 10 cm. Assume h ≠ 0.
A=(h(b_1+b_2))/2
2A=h(b_1+b_2)
(2A)/h=b_1+b_2
b_2=(2A)/h-b_1
b_2=(2(96))/8-10=14
The second base is 14 cm.