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What does (f∘g)(x) mean?
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All Flashcards in Topic 2.7
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2.7.18 cards
What does (f∘g)(x) mean?
f(g(x)): apply the inner function g first, then the outer function f.
In f∘g, which function acts first?
g — the one written closest to x (read right-to-left).
How do you build the formula for (f∘g)(x)?
Substitute the whole expression g(x) into f wherever f's input appears, then simplify.
Is f∘g the same as g∘f?
Usually no — order matters, so they are generally different functions.
f(x)=2x+1, g(x)=x−3. Find (f∘g)(5).
g(5)=2, then f(2)=2(2)+1=5.
f(x)=x², g(x)=x+1. Find (f∘g)(x) and (g∘f)(x).
(f∘g)(x)=(x+1)²; (g∘f)(x)=x²+1. They differ.
What two conditions give the domain of f(g(x))?
x must be allowed into g, AND g(x) must be a legal input for f.
f(x)=√x, g(x)=x−4. Domain of (f∘g)(x)?
√(x−4) needs x−4 ≥ 0, so x ≥ 4.
2.7.28 cards
What does the inverse function f⁻¹ do?
It undoes f: if f maps a→b, then f⁻¹ maps b→a. So f⁻¹(f(x)) = x.
How do you find f⁻¹ algebraically?
Write y = f(x), swap x and y, then solve for y.
What does f⁻¹(b) mean numerically?
The input that gives output b — i.e. solve f(x) = b.
Is f⁻¹ the same as 1/f?
No — f⁻¹ is the inverse function (reverses f); 1/f is the reciprocal.
How are the graphs of f and f⁻¹ related?
f⁻¹ is the reflection of f in the line y = x; each (a,b) becomes (b,a).
When does an inverse function exist?
Only when f is one-to-one (passes a horizontal line test); otherwise restrict the domain.
How do domain and range change for the inverse?
They swap: the range of f becomes the domain of f⁻¹.
Find f⁻¹ for f(x) = 4x + 3.
y = 4x+3 → x = 4y+3 → f⁻¹(x) = (x−3)/4.
Topic 2.7 study notes
Full notes & explanations for Composite & inverse functions (HL only)
Math AI exam skills
Paper structures, command terms & tips
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