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2.10 · Identify from Table
x values: 1,3,5,7 (+2 each). y values: 2,4,8,16 (×2 each). What type of function?
Which is additive? Which is multiplicative?
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2.10 · Identify from Table
x values: 2,4,8,16 (×2 each). y values: 1,3,5,7 (+2 each). What type of function?
Which is additive? Which is multiplicative?
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2.10 · Identify from Table
x values: 5, 50, 500, 5000 (×10). y values: 1, 2, 4, 8 (×2). What type of function?
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2.10 · General Form
Write the general form of a logarithmic function. State the restrictions on a and b.
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2.10 · Inverse Pair
If f(x) = bx and g(x) = logb x, what is f(g(x))? What is g(f(x))?
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2.10 · Apply: Verify
Show that f(x) = 3x and g(x) = log3 x are inverses.
Compute both f(g(x)) and g(f(x))
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2.10 · Graphing Inverse
k(x) = 2x contains (3, 8). What point is on k−1(x) = log2 x?
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2.10 · Graph Facts
State the key graph facts for y = logb x (with b > 1): domain, range, asymptote, key points.
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✓ Logarithmic
Logarithmic — x is multiplicative, y is additive
The inverse of exponential: x-values multiply, y-values add equally.
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✓ Exponential
Exponential — x is additive, y is multiplicative
x adds equally (+2) while y multiplies by a constant (×2) → exponential function.
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✓ General Form
f(x) = a · logb x where b > 0, b ≠ 1, a ≠ 0
b is the base (must be positive, not 1). a is a vertical scaling factor (non-zero).
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✓ Neither
Neither — both x and y are multiplicative
x ×10 and y ×2 — both change multiplicatively. For log: x multiplicative + y additive. For exp: x additive + y multiplicative.
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✓ Verify Inverses
f(g(x)) = 3log₃x = x ✓ g(f(x)) = log3(3x) = x ✓
Both compositions equal x, confirming they are inverses. The base must match.
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✓ Inverse Pair
f(g(x)) = x g(f(x)) = x
bx and logb x undo each other completely — they are inverses. The base b must match.
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✓ Graph Facts
Domain: x > 0 • Range: all reals
VA at x = 0 • Passes through (1,0) and (b,1)
Reflection of bx over y = x. Increasing when b > 1. Decreasing when 0 < b < 1.
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✓ Graphing Inverse
(8, 3)
Inverses swap x and y. (3,8) on 2x → (8,3) on log2 x. The graphs are reflections over y = x.
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2.10 · Apply: AROC
h(x) = ax contains (2,3) and (6,27). Find the AROC of y = loga x on [3, 27].
First: find the points on the inverse
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2.10 · Identify from Table
x: 10, 30, 90, 270 (×3). y: 10, 20, 30, 40 (+10). What type of function?
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2.10 · Identify from Table
x: 4, 8, 16, 32 (×2). y: −1, −4, −7, −10 (−3 each). What type of function?
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2.10 · Connecting Ideas
What is the relationship between the graph of f(x) = bx and the graph of g(x) = logb x?
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✓ Logarithmic
Logarithmic — x multiplies (×3), y adds (+10)
Multiplicative x + additive y = logarithmic function.
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✓ AROC = 1/6
Swap points: (3,2) and (27,6). AROC = (6−2)/(27−3) = 4/24 = 1/6
Inverse swaps x and y: (2,3)→(3,2) and (6,27)→(27,6).
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✓ Connecting Ideas
They are reflections of each other over the line y = x
Because they are inverse functions, all x and y coordinates are swapped, producing a mirror image over y = x.
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✓ Logarithmic
Logarithmic — x multiplies (×2), y adds (−3 each)
The y-values decrease additively (by 3 each step) while x multiplies. Still logarithmic — the a coefficient is negative.