g(x) = 3f(x) compared to f(x) is: — Vertically stretched by factor 3.a·f(x) with |a| > 1 vertically stretches the graph by factor a. All y-values are multiplied by 3.
g(x) = f(2x) compared to f(x) is: — Horizontally compressed by factor 2.f(bx) with |b| > 1 horizontally COMPRESSES by factor b. Points come closer together horizontally.
g(x) = (1/2)f(x) compared to f(x) is: — Vertically compressed by factor 2.a·f(x) with 0 < a < 1 vertically compresses the graph. All y-values are halved.
g(x) = √(x + 4) − 1 is f(x) = √x shifted: — Left 4, down 1.f(x+4) → left 4 (inside, + means left). −1 outside → down 1.
g(x) = |x − 2| + 3 is f(x) = |x| shifted: — Right 2, up 3.f(x−2) → right 2. +3 outside → up 3.
The point (3, 5) on f(x) maps to which point on f(x) + 7? — (3, 12).Vertical shift up 7: y-coordinate increases by 7. New point: (3, 5+7) = (3, 12).
Which parameter in g(x) = a·f(b(x − h)) + k produces vertical stretches? — a.Parameter a (multiplied outside) produces vertical stretches (|a| > 1) or compressions (|a| < 1). It also reflects when a < 0.
Horizontal transformations (inside f) work in the _____ direction of the algebraic sign. — Opposite.This is the key rule: f(x − h) shifts RIGHT (opposite of minus), f(x + h) shifts LEFT (opposite of plus). Inside transformations are counterintuitive.
g(x) = f(x) × 0.25 compared to f(x) is: — Vertically compressed by factor 4.a = 0.25 < 1 → vertical compression. All y-values are multiplied by 0.25 (become 1/4 as tall).
g(x) = √(−x) compared to f(x) = √x is: — Reflected over the y-axis.f(−x) reflects the graph over the y-axis. √(−x) is defined for x ≤ 0 (left side), a mirror of √x.
The vertex of g(x) = (x − 3)² + 5 is at: — (3, 5).g(x) = f(x−3) + 5 shifts the vertex of f(x) = x² (at origin) right 3 and up 5. Vertex: (3, 5).
The graph of f⁻¹ is the reflection of f over the line: — y = x.The inverse function's graph is the reflection of the original over the line y = x. This swaps all (a, b) pairs to (b, a).
f(x) = x³. What is f⁻¹(x)? — x^(1/3) = ∛x.y = x³ → x = y³ → y = x^(1/3) = ∛x. The cube root undoes cubing.
If f(f⁻¹(x)) = x, then applying f⁻¹ then f returns: — x (the original input).f(f⁻¹(x)) = x — applying inverse then original returns the original input x.
f(x) = 5 is a constant function. Does it have an inverse function? — No — it fails the horizontal line test.f(x) = 5 maps every x to 5 — it is many-to-one, failing the horizontal line test. It has no inverse function.
f(x) = x², g(x) = 3. Find (f ∘ g)(x). — 9.(f ∘ g)(x) = f(g(x)) = f(3) = 3² = 9. This is a constant function.
If f and g are inverse functions, the composition graph f(g(x)) = x represents which line? — y = x.f(g(x)) = x is the identity function, represented by the line y = x.
Composition of f followed by g is written as: — g ∘ f.(g ∘ f)(x) = g(f(x)) — f is applied first. The notation g ∘ f reads "g of f" and applies f first.
A piecewise function is CONTINUOUS if: — At each boundary point, both pieces give the same value.Continuity requires that at each boundary/transition point, the function values from both sides match — no "jumps."
f(x) = {2x + 1 if x < 2; 7 if x = 2; x² − 1 if x > 2}. Is f continuous at x = 2? — No — 2(2)+1=5 ≠ 7 at x = 2.Left limit: 2(2)+1 = 5. Right limit: 4−1 = 3. f(2) = 7. All three differ → NOT continuous at x = 2.
Which real-world situation can be modeled by a piecewise function? — A shipping rate that changes based on package weight.Shipping rates use different formulas for different weight ranges — a perfect piecewise model. Tax brackets work the same way.
The graph of a piecewise function can have: — Multiple distinct pieces that may or may not connect.A piecewise function's graph has separate pieces over different x-intervals. They may connect (continuous) or not (discontinuous).
What does an "open circle" on a piecewise graph indicate? — The endpoint is NOT included (the domain condition is strict, < or >).An open circle at a point means that point is NOT included in the piece — a strict inequality (< or >) at that boundary.
What does a "closed circle" on a piecewise graph indicate? — The endpoint IS included (the domain uses ≤ or ≥).A closed (filled) circle indicates the endpoint IS included — a non-strict inequality (≤ or ≥) at that boundary.
For the 3-piece function above, what is f(−3) + f(0) + f(2)? — All three are correct (9).f(−3): x<−1, use −x = 3. f(0): −1≤x<2, use x+2 = 2. f(2): x≥2, use 4. Sum = 3+2+4 = 9.
Which mathematical concept is modeled by f(x) = ⌊x⌋ (the floor function)? — A piecewise constant (step) function.The floor function ⌊x⌋ equals the greatest integer ≤ x. It creates a step pattern — constant on each interval, with jumps at integers.