Is the relation {(1,2), (2,3), (1,4)} a function? — No — input 1 maps to two outputs.Input 1 maps to both 2 and 4 — not a function (violates the "exactly one output" rule).
What is the domain of h(x) = √(x + 5)? — x ≥ −5.Radicand must be ≥ 0: x + 5 ≥ 0 → x ≥ −5.
If f(x) = 2x + 1 and g(x) = x − 3, find (f + g)(x). — 3x − 2.(f+g)(x) = f(x)+g(x) = (2x+1)+(x−3) = 3x−2.
If f(x) = 5, what is the range? — {5}.f(x) = 5 is a constant function. The output is always 5, so range = {5}.
If f(x) = |x|, what is the domain and range? — Domain: all reals, Range: y ≥ 0.|x| is defined for all real x (domain: all reals). The absolute value is never negative (range: y ≥ 0).
Find the equation of the line through (0, 5) with slope 2. — y = 2x + 5.Using slope-intercept: y = mx + b with m=2, b=5 (given point is the y-intercept). y = 2x + 5.
What is the x-intercept of y = 3x − 9? — x = 3.Set y = 0: 0 = 3x − 9 → x = 3.
Write the equation through (2, 5) with slope 3 using point-slope form. — y − 5 = 3(x − 2).Point-slope: y − y₁ = m(x − x₁) → y − 5 = 3(x − 2).
What is the slope of a vertical line? — Undefined.Vertical lines have undefined slope (division by zero in Δy/Δx when Δx = 0).
If line A has slope 2, a perpendicular line has slope: — −1/2.Perpendicular slopes are negative reciprocals: −1/m = −1/2.
Convert 3x − 4y = 12 to slope-intercept form. — y = (3/4)x − 3.−4y = −3x + 12 → y = (3/4)x − 3.
A rate of change of −3 means: — Output decreases by 3 per unit input.Slope = rate of change. −3 means as x increases by 1, y decreases by 3.
Which form makes it easiest to find the slope and y-intercept directly? — Slope-intercept y=mx+b.In y = mx + b, slope m and y-intercept b are immediately visible without any algebra.
For y = 3, 9, 27, 81, … what is the common ratio? — 3.Each term is multiplied by 3. Common ratio = 3. This is a geometric sequence with exponential growth.
The second differences of y = 2, 5, 10, 17, 26 are: — 2, 2, 2.First diffs: 3, 5, 7, 9. Second diffs: 2, 2, 2. Constant second differences → quadratic.
Population doubles every year. Starting at 100, after 4 years: — 1600.100 × 2⁴ = 100 × 16 = 1600.
For f(x) = x² and g(x) = 2ˣ, which has a higher value at x = 10? — g(10) = 1024.g(10) = 2¹⁰ = 1024 > f(10) = 100.
A bacteria culture grows linearly, adding 50 per hour. Starting at 200, after 6 hours? — 500.Linear growth: 200 + 50(6) = 200 + 300 = 500.
Which function type has a constant RATIO of consecutive terms? — Exponential.Exponential: consecutive terms have a constant ratio (common ratio r). Linear has constant differences.
Identify by first differences: x: 0,1,2,3; y: 1, 3, 9, 27. — Exponential.First diffs: 2, 6, 18 — not constant. Ratios: 3/1=3, 9/3=3, 27/9=3 — constant ratio 3. Exponential.
Which function type can model compound interest? — Exponential.Compound interest: A = P(1+r)ⁿ — this is exponential in n (number of periods).
If y = 8 when x = 2 (direct variation), find y when x = 7. — 28.k = 8/2 = 4. y = 4x. When x=7: y = 28.
If y = 6 when x = 3 (inverse variation), find y when x = 9. — 2.k = xy = 6·3 = 18. y = 18/x. When x=9: y = 2.
In an inverse variation, xy = 30. What is y when x = 6? — 5.y = 30/x = 30/6 = 5.
The graph of direct variation y = kx is: — A straight line through origin.y = kx is a linear function (degree 1) passing through the origin.
The graph of inverse variation y = k/x is: — A hyperbola with two branches.y = k/x graphs as a rectangular hyperbola with branches in quadrants 1 and 3 (if k > 0).
Does y = 3x + 1 represent direct variation? — No — direct variation requires y-intercept = 0.Direct variation y = kx has NO constant term. y = 3x + 1 has b = 1 ≠ 0, so it is NOT direct variation.
Earnings and hours worked at a fixed wage show which type? — Direct — earnings = wage × hours.Earnings = k × hours (k = hourly wage). This is direct variation.
In direct variation, the ratio y/x is always: — Equal to k (constant).y = kx → y/x = k. The ratio is always the constant of variation.
In inverse variation, the product xy is always: — Equal to k (constant).y = k/x → xy = k. The product is always the constant of variation.
If y varies directly with x and the graph passes through (4, 20), what is the equation? — y = 5x.k = y/x = 20/4 = 5. Equation: y = 5x.