Functions, domain & range
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What does f(x) mean?
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All Flashcards in Topic 2.2
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2.2.110 cards
What does f(x) mean?
The output of function f for input x. f(3) means substitute x = 3 into the rule.
Is f(x) the same as f × x?
No — it's 'f of x', the function applied to x. The brackets hold the input.
What makes a rule a function?
Each input gives exactly ONE output. (Different inputs may share an output.)
How do you evaluate f(a)?
Replace every x with a (in brackets for negatives/expressions), then simplify.
Evaluate g(x) = x² − 4x at x = −3.
(−3)² − 4(−3) = 9 + 12 = 21.
How do you solve f(x) = k?
Set the rule equal to k and solve for x (output → input).
Can two inputs give the same output?
Yes — e.g. f(x) = x² gives f(2) = f(−2) = 4. So f(x) = k may have several solutions.
How do you read f(a) off a graph?
Go up from x = a to the curve, then across to the y-axis.
How do you solve f(x) = k off a graph?
Read across from y = k to the curve, then down to the x-axis (there may be several x).
Find f(2a) for f(x) = 3x − 5.
Substitute the whole expression: 3(2a) − 5 = 6a − 5.
2.2.210 cards
What is the domain of a function?
The set of all allowed inputs (x-values).
What is the range of a function?
The set of all possible outputs (y-values).
How do you read the domain off a graph?
How far the graph extends left ↔ right (the x-extent).
How do you read the range off a graph?
How far the graph extends down ↕ up (the y-extent).
What two things restrict a domain?
No dividing by zero (denominator ≠ 0) and no even root of a negative (under √ ≥ 0). Also a log argument must be > 0.
Domain of 1/(x − 3)?
x ≠ 3 — the denominator can't be zero.
Domain of √(x − 2)?
x ≥ 2 — what's under the root must be ≥ 0 (0 is allowed).
Range of f(x) = (x − h)² + k opening upward?
y ≥ k — the vertex (h, k) is the minimum.
Range of an exponential aˣ (a > 0)?
y > 0 — always positive, approaching but never reaching 0.
What's the default domain if nothing restricts it?
All real numbers, x ∈ ℝ.
2.2.310 cards
What does an inverse function do?
It undoes f: if f(a) = b then f⁻¹(b) = a. Inputs and outputs swap.
Is f⁻¹(x) the same as 1/f(x)?
No — f⁻¹ is the inverse function (reverses f), not the reciprocal.
The graph of f⁻¹ is f reflected in which line?
y = x. Each point (a, b) on f becomes (b, a) on f⁻¹.
How do you find f⁻¹ algebraically?
Write y = f(x), swap x and y, then solve for y — that's f⁻¹(x).
Find the inverse of f(x) = 2x + 3.
Swap: x = 2y + 3 ⇒ y = (x − 3)/2, so f⁻¹(x) = (x − 3)/2.
How do domain and range change for f⁻¹?
They swap: domain of f⁻¹ = range of f; range of f⁻¹ = domain of f.
Where do f and f⁻¹ intersect?
On the line y = x — solve f(x) = x to find where.
How can you check an inverse?
Pick a point: f(a) = b should give f⁻¹(b) = a. Or check f(f⁻¹(x)) = x.
Why might f⁻¹ need a restricted domain?
Its domain is f's range, which can be limited (e.g. √x has range y ≥ 0, so its inverse x² is restricted to x ≥ 0).
What happens to a point already on y = x under reflection?
It maps to itself — which is why f and f⁻¹ meet on y = x.
Topic 2.2 study notes
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Math AA SL exam skills
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