Functions & Notation
If f(x) = 2x + 3, find f(4).
f(4) = 2(4)+3 = 11.
Factor completely: x^2 - 9.
x^2 - 9 = (x - 3)(x + 3).
Simplify: (x^2 - 4)/(x - 2) for x ≠ 2.
(x^2 - 4)/(x - 2) = x + 2 for x ≠ 2.
Solve: 2^3 = ?
2^3 = 8.
Convert 90° to radians.
90° = π/2.
Let f(x)=x^2 and g(x)=x+1. Find (f+g)(2).
(f+g)(2) = f(2)+g(2) = 4 + 3 = 7.
Multiply: (x + 2)(x - 5).
(x + 2)(x - 5) = x^2 - 3x - 10.
Solve: (x/3) + 2 = 5.
x/3 + 2 = 5 → x/3 = 3 → x = 9.
Solve: 10^2 = ?
10^2 = 100.
Evaluate sin 0 and cos 0.
sin 0 = 0; cos 0 = 1.
For f(x) = 1/(x-2), list values of x that are NOT in the domain.
Domain excludes x = 2.
Factor: x^2 + 5x + 6.
x^2 + 5x + 6 = (x + 2)(x + 3).
Solve: sqrt(x + 3) = 4.
sqrt(x + 3) = 4 → x + 3 = 16 → x = 13.
Solve for x: 3^x = 27.
3^x = 27 → x = 3.
Using a 3-4-5 right triangle, find sin θ if opposite = 3 and hypotenuse = 5.
sin θ = opposite/hypotenuse = 3/5.
Simplify the expression (for x ≠ 1): (x^2 - 1)/(x - 1).
(x^2 - 1)/(x - 1) = x + 1 for x ≠ 1.
Divide: (x^3 - x^2) ÷ x and simplify.
(x^3 - x^2)/x = x^2 - x.
Simplify: (6x)/(3x) for x ≠ 0.
(6x)/(3x) = 2 for x ≠ 0.
If P(t) = 500(2)^t, find P(0) and P(1).
P(0) = 500(2^0)=500(1)=500. P(1)=500(2)=1000.
Solve for θ in [0, 2π): sin θ = 0.
sin θ = 0 → θ = 0, π, 2π (within [0,2π) take 0 and π).
If f(x) = 3x - 5 and f(a) = 16, find a.
3a - 5 = 16 → 3a = 21 → a = 7.
Given that x = 2 is a root of p(x) = x^3 + ax^2 + bx + c and p(0)= -8, construct one possible cubic with integer coefficients and leading coefficient 1 that has x=2 as a root and p(0) = -8.
One possible cubic: (x - 2)(x^2 + 1) = x^3 - 2x^2 + x - 2. Adjust constant to get p(0) = -8: multiply or pick (x - 2)(x^2 + x + 4) = x^3 - x^2 + 2x - 8 (this has p(0) = -8 and root 2).
Solve and check for extraneous solutions: sqrt(2x + 1) = 3.
sqrt(2x + 1) = 3 → 2x + 1 = 9 → 2x = 8 → x = 4 (check: sqrt(9)=3 OK).
Solve for x: 5^{x} = 125.
5^x = 125 = 5^3 → x = 3.
Given tan θ = 1 and θ in [0, 2π), list all θ that satisfy this.
tan θ = 1 → θ = π/4 and 5π/4 (within [0,2π)).