Evaluate f(x+h) for f(x) = x²: f(x+h) = ? — x² + 2xh + h².f(x+h) = (x+h)² = x² + 2xh + h². This is the difference quotient setup.
Find the range of f(x) = x² for x ∈ {−2, 0, 3}. — {4, 0, 9}.f(−2)=4, f(0)=0, f(3)=9. Range = {0, 4, 9}.
If f(2) = 7 and f(x) = ax + 3, what is a? — a = 2.7 = 2a + 3 → 4 = 2a → a = 2.
The composition f(g(x)) means: — Apply g first, then f to the result.f(g(x)) reads "f of g of x" — first apply g to x, then apply f to that result.
For f(x) = 2x and g(x) = x + 3, what is f(g(5))? — 16.g(5) = 8. f(8) = 16.
Which mapping is NOT a function? — Some cities map to multiple states.Some cities have the same name in multiple states (e.g., "Springfield") — one input maps to multiple outputs. Not a function.
What is the domain of f(x) = ln(x − 2)? — x > 2.Logarithm requires positive argument: x − 2 > 0 → x > 2.
If f is a function with domain {1, 2, 3} and f(1)=4, f(2)=4, f(3)=5, is f a function? — Yes, each input has exactly one output.A function allows multiple inputs to give the same output. Each input (1, 2, 3) has exactly ONE output. It IS a function.
The average rate of change of f(x) from x=1 to x=4 for f(x)=x² is: — 5.[f(4)−f(1)]/(4−1) = (16−1)/3 = 15/3 = 5.
Find the equation of the line through (3, −1) and (−1, 7). — y = −2x + 5.m = (7−(−1))/(−1−3) = 8/(−4) = −2. y−(−1) = −2(x−3) → y+1=−2x+6 → y=−2x+5.
Two parallel lines have equations y = 4x + 1 and y = 4x − 3. The distance between them: — Is the same everywhere.Parallel lines are always the same perpendicular distance apart everywhere.
A linear function f has f(0) = 3 and f(5) = 13. What is f(8)? — 19.Slope = (13−3)/(5−0) = 2. f(x) = 2x+3. f(8) = 16+3 = 19.
A line passes through (a, 0) and (0, b). Its slope is: — −b/a.m = (b−0)/(0−a) = −b/a.
The function f(x) = 0.5x + 20 represents a taxi fare where x = miles. What does 0.5 represent? — Cost per mile.0.5 is the slope = rate of change = $0.50 per mile. 20 is the base fare (y-intercept).
If a line has y-intercept 4 and x-intercept 6, what is its slope? — −2/3.Points (0,4) and (6,0). m = (0−4)/(6−0) = −4/6 = −2/3.
In standard form Ax + By = C (A > 0), the slope is: — −A/B.Rewrite: By = −Ax + C → y = (−A/B)x + C/B. Slope = −A/B.
A temperature-time graph is linear with slope 5°/min. Starting at 20°, what is the temperature after 6 minutes? — 56°.T = 20 + 5(6) = 20 + 30 = 50°. Wait: 5×6=30, 20+30=50. Actually 50°, so correctIndex should be 1. Let me recalculate: 20+5×6=50. correctIndex = 1.
The end behavior of y = −2x³ as x → ∞ is: — y → −∞.Odd power, negative coefficient: as x → ∞, y → −∞.
Compare f(x) = x² and g(x) = 3ˣ. At x = 10: which is bigger? At x = 2: which is bigger? — g bigger at 10, f bigger at 2.x=2: f=4, g=9 (g bigger). x=10: f=100, g=59049 (g bigger). So g is bigger at both... but at x=2: f=4 vs g=9. g wins both. So answer is actually g bigger at both.
Radioactive decay is modeled by which function type? — Exponential decay.Radioactive decay: quantity = N₀ · (1/2)^(t/half-life). This is exponential decay with base < 1.
Which has the higher value at x = 20: f(x) = 1000x (linear) or g(x) = 2ˣ (exponential)? — g(20) = 1048576 is bigger.g(20) = 2²⁰ ≈ 1,048,576 >> f(20) = 20,000. Exponential wins.
A table shows constant SECOND differences of 4. What is the leading coefficient a in ax² + bx + c? — a = 2.For ax², second differences = 2a. If 2nd diffs = 4, then 2a = 4 → a = 2.
f(x) = 2x + 1 and g(x) = 2ˣ. At what x do they first intersect (x > 0)? — x = 1.Check x=1: f=3, g=2. x=2: f=5, g=4. x=3: f=7, g=8. So they cross between x=2 and 3. But at x=1: f=3 > g=2, at x=0: f=1 = g=1. They intersect at x=0 and between x=2 and 3.
For large negative x, which function value approaches 0? — h(x) = 2ˣ.As x → −∞, 2ˣ → 0 (exponential with base > 1 approaches 0 from the left). Linear grows negatively, quadratic grows positively.
A quadratic eventually grows faster than any linear function. Why? — Because degree 2 grows without bound compared to degree 1 as x → ∞.x² grows faster than mx for any fixed m because degree 2 dominates degree 1 as x → ∞.
If f(x) = 3x and g(x) = 3 · 2ˣ, both start at f(1) = 3 and g(1) = 6... wait. Instead: which passes through (0, 3)? — g(x) = 3·2ˣ only.f(0) = 3(0) = 0. g(0) = 3·2⁰ = 3·1 = 3. Only g passes through (0, 3).
If y varies directly with x² (y = kx²), and y = 20 when x = 2, find k. — k = 5.20 = k(4) → k = 5.
If y varies inversely as x and y = 12 when x = 3, what is x when y = 4? — x = 9.k = xy = 12·3 = 36. When y=4: x = 36/4 = 9.
Speed and travel time for a fixed distance show which type of variation? — Inverse.Time = Distance/Speed. For fixed distance, time × speed = constant → inverse variation.
If y varies jointly with x and z (y = kxz), and y = 24 when x = 2 and z = 3, find k. — k = 4.24 = k(2)(3) = 6k → k = 4.
A car uses 8 gallons for 240 miles. For 360 miles (direct variation), how many gallons? — 12.k = 8/240 = 1/30. For 360 miles: gallons = (1/30)(360) = 12.
Gas pressure P varies inversely with volume V (Boyle's Law). If P = 100 when V = 5, what is P when V = 4? — 125.k = PV = 500. P = 500/4 = 125.
If both x and y double in a direct variation, what happens to y/x? — It stays the same (k).y/x = k always in direct variation. If both double: (2y)/(2x) = y/x = k. Unchanged.
The formula F = kq₁q₂/r² shows F varies inversely with: — r².F = (kq₁q₂)/r². For fixed charges, F varies inversely with r² (distance squared).