a) y = x/5 → linear function
b) y = 1 - [4x - 7 - (1 - 2x) + 3] - x → simplify: y = 1 - [4x - 7 - 1 + 2x + 3] - x = 1 - [6x - 5] - x = 1 - 6x + 5 - x = 6 - 7x → linear function
c) y = (2x+3)/(3x+5) → rational function
d) y = (x-1)/(6x+3) → rational function
e) y = 1 - 1/(2x) → rational function
f) y = (-x-7)/(x+5) → rational function
g) y = (9-3x)/(9x-3) → rational function
h) y = (10x-5)/(15-10x) + 1 → simplify: y = (10x-5)/(15-10x) + 1 = -(10x-5)/(10x-15) + 1 = -[5(2x-1)]/[5(2x-3)] + 1 = -(2x-1)/(2x-3) + 1 = [-(2x-1) + (2x-3)] / (2x-3) = (-2x+1+2x-3)/(2x-3) = (-2)/(2x-3) → rational function
i) y = {1 - [10 - (7 - x) + 20] - 5x} / {1 + 2x - (3 - 4x)} - 2 → simplify numerator: 1 - [10 - 7 + x + 20] - 5x = 1 - [23 + x] - 5x = 1 - 23 - x - 5x = -22 - 6x; denominator: 1 + 2x - 3 + 4x = 6x - 2; so y = (-22 - 6x)/(6x - 2) - 2 = [-22 - 6x - 2(6x - 2)] / (6x - 2) = [-22 - 6x - 12x + 4] / (6x - 2) = (-18 - 18x)/(6x - 2) = -18(1+x)/[2(3x-1)] = -9(1+x)/(3x-1) → rational function
j) y = (x³ - 1) : x³ → y = (x³ - 1)/x³ = 1 - 1/x³ → rational function
k) y = -x² - (-x)² → y = -x² - x² = -2x² → quadratic function
l) y = 2x^(4/3) - 1 → power function
m) y = x³ + 3x² + 3x + 1 → polynomial function (cubic)
n) y = √(x - 1) → square root function (defined for x ≥ 1)
o) y = √[(2x+4)³ - 7] → square root of cubic expression (defined where (2x+4)³ ≥ 7)
p) y = (5 - √(x+2))⁴ + 3 → composition involving square root and power
q) y = 2^x → exponential function
r) y = (1/8)^(1-x) → exponential function (since (1/8)^(1-x) = 8^(x-1))
s) y = -3 × 5^x + 6 → exponential function
t) y = 1 + log x → logarithmic function (defined for x > 0)
u) y = -2 log[(x-1)/(x+1)]⁵ → logarithmic function (using log(a^b)=b log a, so y = -10 log[(x-1)/(x+1)]; defined where (x-1)/(x+1) > 0, i.e., x < -1 or x > 1)
v) y = log x - log 2x + log 3x → using log properties: y = log(x / (2x)) + log(3x) = log(1/2) + log(3x) = log(3x/2) → logarithmic function (defined for x > 0)
w) y = sin 2x + 1 → trigonometric function
x) y = (1 - cos(x/2))² - 1 → trigonometric function
y) y = 2 tan²(x + π/2) - 8 → trigonometric function (note: tan(x + π/2) is undefined at odd multiples of π/2, so domain excludes those points)
z) y = (cos²x - sin²x) / (2 sin x cos x) → use identities: numerator = cos(2x), denominator = sin(2x), so y = cos(2x)/sin(2x) = cot(2x) → trigonometric function
Parent Tip: Review the logic above to help your child master the concept of inverse function word problems worksheet.
