Algebra Review
Solve: |x − 2| ≤ 5. Write your answer in interval notation.
[-3, 7]
Let f(x) = x² and g(x) = x + 3. Find (f ∘ g)(2) and (g ∘ f)(2).
(f∘g)(2) = 25
(g∘f)(2) = 7
Let f(x) = 3x + 5. Find f⁻¹(x).
f⁻¹(x) = (x − 5)/3
A quantity starts at 100 and decreases by 20% each unit of time. Write the exponential model E(t).
E(t) = 100(0.8)ᵗ
$1000 earns an APR of 6%, compounded annually. Find the interest rate and growth factor per period.
Rate = 6% = 0.06 per period
Growth factor = 1.06
Solve the system:
4x + y = 9
2x − y = 3
x = 2, y = 1
Let f(x) = x² and q(x) = 2f(x − 3) − 1.
List, in order, the transformations that take y = f(x) to y = q(x).
Shift right 3, vertical stretch by factor 2, shift down 1
For f(x) = 3x + 5, evaluate f⁻¹(11) and interpret it as solving an equation.
f⁻¹(11) = 2
Solves 3x + 5 = 11
Simplify the ratio E(t + 3)/E(t) for E(t) = 60(0.75)ᵗ, and interpret what it represents.
Ratio = 0.75³ = 27/64
Represents the multiplying change over any 3-period span
$2000 earns an APR of 12%, compounded monthly. Write an exact expression for the value after 2 years.
A = 2000(1.01)²⁴
Solve exactly for all real x:
√(2x² + 7) = x + 2
(Check for extraneous solutions.)
x = 1
x = 3
A shipping process adds 2 inches of packaging, P(x) = x + 2, then applies a 10% size buffer, L(x) = 1.1x. If the final dimension L(P(x)) must equal 33 inches, find the original size x.
x = 28
(1.1(x+2) = 33 → x + 2 = 30 → x = 28)
Let f(x) = (x + 3)/(x − 2). Find f⁻¹(x), showing the algebra, and state any domain restriction.
f⁻¹(x) = (2x + 3)/(x − 1)
x ≠ 1
A population starts at 200 and grows 50% every 2 years. Write P(t) in terms of years t, and evaluate P(6).
P(t) = 200(1.5)^(t/2)
P(6) = 675
Two accounts start with the same principal P, both at 8% APR. Account A compounds quarterly, Account B compounds monthly. Write exact expressions for both after t years and explain why B grows faster.
A: P(1 + 0.08/4)^(4t)
B: P(1 + 0.08/12)^(12t)
More frequent compounding → higher effective annual rate, so B grows faster